Variation bounds for spherical averages
David Beltran, Richard Oberlin, Luz Roncal, Andreas Seeger, Betsy Stovall
VVARIATION BOUNDS FOR SPHERICAL AVERAGES
DAVID BELTRAN RICHARD OBERLIN LUZ RONCALANDREAS SEEGER BETSY STOVALL
Abstract.
We consider r -variation operators for the family of sphericalmeans, with special emphasis on L p → L q estimates.
1. Introduction
Given a subset E ⊂ R and a family of complex valued functions t (cid:55)→ a t defined on E , the r -variation of a = { a t } t ∈ E is defined by | a | V r ( E ) := sup N ∈ N sup t < ···
3) and Bourgain [7] ( d = 2) thespherical maximal function Sf ( x ) := sup t> | A t f ( x ) | defines a bounded op-erator on L p ( R d ) if and only if p > dd − . Thus, for p in this range, we havelim t → A t f ( x ) = f ( x ) a.e. for all f ∈ L p ( R d ). A strengthening of this resultcan be obtained by considering the variation norm operator V r A given by V r Af ( x ) ≡ V r [ Af ]( x ) := | Af ( x ) | V r ((0 , ∞ )) ; Date : September 17, 2020, Preliminary manuscript.2010
Mathematics Subject Classification.
Primary 42B15, 42B25.
Key words and phrases. spherical averages, variation norm, L p → L q estimates. a r X i v : . [ m a t h . C A ] S e p D. BELTRAN, R. OBERLIN, L. RONCAL, A. SEEGER, B. STOVALL note that V r [ Af ]( x ) ≥ sup t | A t f ( x ) − A t f ( x ) | for all x ∈ R d , t ∈ R . Inthis context, Jones, Wright and one of the authors [21] obtained an almostoptimal result, namely V r A is bounded on L p ( R d ) for all r > dd − < p ≤ d , and both the condition r > p -range are sharp. In the range p > d , it was shown in [21] that V r A is L p bounded if r > p/d , and fails tobe bounded if r < p/d , but no information was known for the critical case r = p/d , p > d . Here we show an endpoint result for V p/d A in three andhigher dimensions. Theorem 1.1.
Let d ≥ , p > d . Then the operator V p/d A is of restrictedweak type ( p, p ) , i.e. maps L p, ( R d ) to L p, ∞ ( R d ) . We conjecture that a similar endpoint result holds true in two dimensions,but this remains open.Our main focus will be on L p → L q results when p < q for local r -variationoperators, that is, when the variation is taken over a compact subinterval I of(0 , ∞ ); without loss of generality we take I = [1 , L p → L q boundsto hold if p < q . While this is an interesting problem in itself, it is alsomotivated by a question posed by Lacey [23] concerning sparse dominationfor the global V r A operator (see also [1, Problem 3.1]). See Theorem 1.7below.Results for the local variation operators are meant to improve on ex-isting L p → L q results for the spherical local maximal function S I f ( x ) :=sup ≤ t ≤ A t f ( x ), which we will now review. Schlag [37] (see also [38]) showedthat if d ≥ L p ( R d ) → L q ( R d ) bounds if (1 /p, /q ) lies in the in-terior of Q d , which denotes the quadrangle formed by the vertices Q = (0 , , Q = ( d − d , d − d ) ,Q = ( d − d , d ) , Q = ( d ( d − d +1 , d − d +1 ) . (1.1)Moreover, S I fails to be bounded from L p ( R d ) to L q ( R d ) outside theclosure of Q d . Note that Q coincides with Q when d = 2, so the quadranglebecomes a triangle in two dimensions.The boundary segment p = q amounts to the classical results of Steinand Bourgain for S . L p -boundedness fails at the endpoint Q but Bour-gain showed in dimensions d ≥ S is of restricted weak type at Q ,i.e. bounded from L dd − , to L dd − , ∞ in dimensions d ≥ Q fails in twodimensions [40] (even though it is true for radial functions [24]). For theremaining boundary cases Lee [25] showed that S I is of restricted weak typeat Q , and also at Q in dimensions d ≥
3. The two-dimensional restrictedweak type endpoint result at Q was also shown in [25], and relied on thedeep work by Tao [45] on endpoint bilinear Fourier extension bounds for thecone. The restricted weak type inequalities imply L p → L q boundedness on[ Q , Q ) and on ( Q , Q ), however on ( Q , Q ) the operator is of restricted ARIATION BOUNDS FOR SPHERICAL AVERAGES 3 strong type and no better (the necessity follows from the standard coun-terexample; for the positive result one uses real interpolation on a verticalline, with a constant target exponent). Incidentally, for the local operator S I this also implies restricted strong type at Q , which improves over therestricted weak type of S at Q .Here we explore the existence of L p ( R d ) → L q ( R d ) inequalities for V Ir Af ( x ) := | Af ( x ) | V r ([1 , . In two dimensions the values of r are restricted to r > §
3) but in higherdimensions all r ∈ [1 , ∞ ] may occur. For our sparse domination inequalityfor the global V r , the version for r > r > V r .We start stating our results for d ≥
3. We first focus on the range r > d +1 d ( d − which is the reciprocal of the 1 /p coordinate of the point Q in (1.1).Note that this large range includes r >
2, so the following sharp L p → L q results for V Ir A will yield, in particular, satisfactory results for the sparsedomination problem in dimension d ≥ Theorem 1.2.
Suppose d ≥ and r > d +1 d ( d − . Let P d ( r ) be the pentagon(Figure 1) with vertices P ( r ) = ( r , rd ) , Q ( r ) = ( rd , rd ) , Q = ( d − d , d − d ) Q = ( d − d , d ) , Q = ( d ( d − d +1 , d − d +1 ) . Then(i) V Ir A : L p → L q is bounded for all ( p , q ) in the interior of P d ( r ) andunbounded for all ( p , q ) / ∈ P d ( r ) .(ii) V Ir A : L p → L q is bounded for all ( p , q ) on the half open line segment [ Q ( r ) , Q ) , on the closed line segment [ P ( r ) , Q ( r )] , on the half open linesegment [ P ( r ) , Q ) , and on the open line segment ( Q , Q ) .(iii) V Ir A : L p, → L q is bounded (i.e. of restricted strong type ( p, q ) )if ( p , q ) belongs to the half open line segment [ Q , Q ) . V Ir A fails to be ofstrong type on [ Q , Q ] .(iv) V Ir A : L p, → L q, ∞ is bounded (i.e. of restricted weak type ( p, q ) ) if ( p , q ) ∈ { Q , Q } . For an explicit description of the various conditions at the boundary see § Q and Q ; it is noteven known whether the local maximal function is of restricted strong typeat Q and whether it is any better than restricted weak type at Q . If wetake r = ∞ we recover the known theorem for the local spherical maximaloperator. Note that both P ( r ) and Q ( r ) tend to Q = (0 ,
0) as r → ∞ . D. BELTRAN, R. OBERLIN, L. RONCAL, A. SEEGER, B. STOVALL1 p q P ( r ) r Q ( r ) rdd − d Q d − d Q Q d − d d Figure 1.
The pentagon P d ( r ) for r > d +1 d − d and d ≥ r → ∞ , i.e. for the maximal operator.Shown with d = 4 and r = 3.Theorem 1.2 covers an interesting consequence for a sharp strong typeestimate at the lower edge q − = p − /d of the type set for the maximalfunction. Corollary 1.3.
Let d ≥ and let d +1 d ( d − < p < ∞ . Then V Ir A : L p → L pd is bounded if and only if r ≥ p . When the value of r is between the reciprocal of the 1 /p coordinate of Q and Q , that is, dd − < r ≤ d +1 d ( d − , we obtain the following. Theorem 1.4.
Suppose d ≥ and dd − < r ≤ d +1 d ( d − . Let P d ( r ) be thepentagon (Figure 2) with vertices Q ( r ) = (cid:0) rd , rd (cid:1) , Q = (cid:0) d − d , d − d (cid:1) , Q = (cid:0) d − d , d (cid:1) P ( r ) = (cid:0) r , d +1 − r ( d − r ( d − (cid:1) , Q ( r ) = (1 − d +1 rd ( d − , rd ) . Then(i) V Ir A : L p → L q is bounded for ( p , q ) in the interior of P d ( r ) andunbounded for ( p , q ) / ∈ P d ( r ) .(ii) V Ir A : L p → L q is bounded for ( p , q ) on the half open line segment ( Q ( r ) , Q ( r )] and on the half open line segment [ Q ( r ) , Q ) .(iii) V Ir A is of restricted strong type ( p, q ) if ( p , q ) belongs to the half openline segment [ Q , Q ) . V Ir A fails to be of strong type on [ Q , Q ] .(iv) V Ir A is of restricted weak type ( p, q ) if ( p , q ) = Q . ARIATION BOUNDS FOR SPHERICAL AVERAGES 5
Note that for r = d +1 d ( d − the pentagon P d ( r ) in Figure 2 degenerates toa quadrangle, as P ( r ) = Q ( r ) = Q . We leave open what happens at theclosed boundary segment [ Q ( r ) , P ( r )] and the half-open boundary segment[ P ( r ) , Q ). p q
12 12 P ( r ) Q ( r ) d − d Q ( r ) d − d Q Q d − d d rd Figure 2.
The pentagon P d ( r ) for dd − < r ≤ d +1 d − d and d ≥ d = 4 and r = .Finally, we address small values of r . Theorem 1.5.
Suppose that either d ≥ and ≤ r ≤ dd − or d = 3 and < r ≤ . Let Q d ( r ) be the quadrangle (Figure 3) with vertices Q ( r ) = (cid:0) rd , rd (cid:1) , Q ( r ) = (cid:0) r ( d − − r ( d − , r ( d − − r ( d − ) ,Q ( r ) = ( r ( d − − r ( d − , r ( d − ) , Q ( r ) = (1 − d +1 rd ( d − , rd ) . Then(i) V Ir A : L p → L q is bounded for ( p , q ) in the interior of Q d ( r ) andunbounded for ( p , q ) / ∈ Q d ( r ) .(ii) V Ir A : L p → L q is bounded if ( p , q ) is in the half open line segment ( Q ( r ) , Q ( r )] and [ Q ( r ) , Q ( r )) .(iii) For the case r = 1 , d ≥ , the operator V I A is of restricted weaktype ( d − d − , d − (that is, at Q (1) ) and of restricted strong type ( d − d − , q ) for d − d − ≤ q < d − (that is, on [ Q (1) , Q (1)) ). In three dimensions, V I A : L ( R ) → L ( R ) is bounded. We leave open what happens at the closed boundary segments [ Q ( r ) , Q ( r )]for 1 < r ≤ dd − and [ Q ( r ) , Q ( r )] for 1 ≤ r ≤ dd − . D. BELTRAN, R. OBERLIN, L. RONCAL, A. SEEGER, B. STOVALL1 p q Q ( r ) Q ( r ) Q ( r ) d − d Q ( r ) d − d d − d d rd Figure 3.
The quadrangle Q d ( r ) for 1 ≤ r ≤ dd − and d ≥ d = 4and r = . Figure 4.
A diagram of the typeset of V Ir A in ( p , q , r )-space for large values of d . The green region correspondsto Theorem 1.2 (Figure 1), the red region corresponds toTheorem 1.4 (Figure 2), and the blue region corresponds toTheorem 1.5 (Figure 3). The yellow region is conjectural. ARIATION BOUNDS FOR SPHERICAL AVERAGES 7
Figure 5.
A diagram of the typeset of V Ir A in ( p , q , r )-space for d = 3. The green region corresponds to Theorem1.2 (Figure 1), the red region corresponds to Theorem 1.4(Figure 2), and the blue region corresponds to Theorem 1.5(Figure 3). The yellow region is conjectural.Note that there is a discrepancy in our results between d = 3, for whichwe only obtain sharp results in the partial range < r ≤ dd − and the case d ≥
4, where results are obtained for all 1 ≤ r ≤ dd − . The reason is becausewe restrict ourselves to the traditional range 1 ≤ r ≤ ∞ for the variationnorm. The definition of V r can be extended, with modifications, to therange 0 < r < V Ir A for d − < r < d ≥
4. Weremark that a positive solution to Sogge’s local smoothing conjecture [41]in d + 1 dimensions would imply a complete result up to endpoints. Partialresults in the range r > d +1) d ( d − can be proved using the techniques of thispaper. We shall address issues for r < ≤ r ≤ / V r spaces.In dimension 2, due to the recent full resolution of Sogge’s problem in2 + 1 dimensions by Guth, Wang and Zhang [17], that is, ∂ / − εt A : L → L ( L ) , it is possible to get an almost optimal result (up to endpoints) for the vari-ation norm estimates. D. BELTRAN, R. OBERLIN, L. RONCAL, A. SEEGER, B. STOVALL
Theorem 1.6.
Let d = 2 .(i) If r > / then V Ir A : L p → L q is bounded if ( p , q ) is either in theinterior of the quadrangle Q ( r ) (Figure 6) formed by the vertices P ( r ) = ( r , r ) , Q ( r ) = ( r , r ) ,Q = Q = ( , ) , Q = ( , ) or in the open line segment between Q = Q and Q ( r ) .(ii) If < r ≤ / then V Ir A : L p → L q is bounded if ( p , q ) is either inthe interior of the quadrangle Q ( r ) (Figure 7) formed by the vertices Q ( r ) = (cid:0) r , r (cid:1) , Q = Q = (cid:0) , ) ,P ( r ) = ( r , − rr ) , Q ( r ) = (1 − r , r ) or in the open line segment between Q = Q and Q ( r ) .(iii) If r < then V Ir A does not map any L p ( R ) to any L q ( R ) . p q Q Q = Q r Q ( r ) P ( r ) r Figure 6.
The region Q ( r ) if r > / r = 5.Note that, as for the circular maximal function theorem, the points Q and Q coincide if d = 2; therefore the pentagon (Figures 1 and 2) in Theorems1.2 and 1.4 becomes a quadrangle for r >
2. Moreover, P (5 /
2) = Q (5 /
2) = Q , so the quadrangle becomes a triangle for r = 5 /
2. The bounds aresubsumed in Figure 8; note that in contrast with d ≥
3, the blue/yellowregion disappears, as dd − = d − coincide for d = 2.It is also possible to show unboundedness for r = 2 via an argument in-volving the Besicovitch set, which will be addressed in a forthcoming paper.We note that an affirmative answer to endpoint versions of Sogge’s prob-lem as formulated and conjectured in [18] would also settle strong typebounds on the half-open boundary segment ( Q , Q ( r )]. Unfortunately such ARIATION BOUNDS FOR SPHERICAL AVERAGES 9 endpoint bounds in Sogge’s problem are currently only available in dimen-sions four and higher. p q r r r − r − rr Q = Q Q ( r ) Q ( r ) P ( r ) Figure 7.
The region Q ( r ) if d = 2 and 2 < r ≤ / r =2 . Figure 8.
A diagram of the typeset of V Ir A in ( p , q , r )-space for d = 2. The green region corresponds to Figure 6and the red region corresponds to Figure 7. Sparse domination.
We now formulate a sparse domination result for theglobal operator V r A , r >
2. Recall that a family of cubes S in R d is called sparse if for every Q ∈ S there is a measurable subset E Q ⊂ Q such that | E Q | ≥ | Q | / { E Q : Q ∈ S } are pairwisedisjoint. In what follows we abbreviate (cid:104) f (cid:105) Q,s = ( | Q | − (cid:82) Q | f | s ) /s . Theorem 1.7.
Assume one of the following holds:(i) d ≥ , r > , and ( p , q ) in the interior of P d ( r ) .(ii) d = 2 , r > and ( p , q ) in the interior of Q ( r ) .Then there is a constant C = C ( p, q ) such that for each pair of compactlysupported bounded functions f , f there is a sparse family of cubes S suchthat (cid:90) R d V r Af ( x ) f ( x ) d x ≤ C (cid:88) Q ∈ S | Q |(cid:104) f (cid:105) Q,p (cid:104) f (cid:105) Q,q (cid:48) , (1.2) where q + q (cid:48) = 1 . Furthermore, the (1 /p, /q ) range is sharp up to endpointsin the sense that no such result can hold if (1 /p, /q ) does not lie in theclosure of P d ( r ) , or Q ( r ) , respectively. Theorem 1.7 can be obtained as an immediate consequence of a (moregeneral) sparse domination result in [2], together with the L p results in [21]and Theorems 1.2 and 1.6; see § §
8. Sparse domination is knownto imply as a corollary a number of weighted inequalities in the context ofMuckenhoupt and reverse H¨older classes. We refer the interested reader to[5] for the weighted consequences for V r A of Theorem 1.7. Structure of the paper.
We start gathering some well known facts aboutspherical averages and function spaces in §
2. In § § §§ § § Acknowledgements.
We are indebted to Shaoming Guo for useful contribu-tions at various stages of the project. Some initial work on this project wasdone during the workshop “Sparse domination of singular integral opera-tors” in October 2017, attended by four of the authors. We would like tothank the American Institute of Mathematics for hosting the workshop, aswell as the organizers Amalia Culiuc, Francesco Di Plinio and Yumeng Ou.D.B. was partially supported by the NSF grant DMS-1954479. L.R. waspartially supported by the Basque Government through the BERC 2018-2021 program, by the Spanish Ministry of Economy and Competitiveness:BCAM Severo Ochoa excellence accreditation SEV-2017-2018 and through
ARIATION BOUNDS FOR SPHERICAL AVERAGES 11 project MTM2017-82160-C2-1-P, by the project RYC2018-025477-I, and byIkerbasque. A.S. was partially supported by NSF grant DMS-1764295 andby a Simons fellowship. B.S. was partially supported by NSF grant DMS-1653264.
2. Preliminaries
It will be convenient to consider the t -parameter as a variable. To thisend, let χ ∈ C ∞ c ( R ) so that χ ( t ) = 1 for t in a neighborhood of [1 ,
2] andsupported in [1 / , A f ( x, t ) := χ ( t ) A t f ( x ) . (2.1)In view of future frequency decompositions, let β ∈ C ∞ c ( R ) so that β ( s ) =1 for | s | < / β ( s ) = 0 for | s | >
1. For every integer j ≥
1, set β j ( s ) = β (2 − j s ) − β (2 − j s ) . For functions g on R , and l ∈ N , define the operators Λ l by (cid:100) Λ l g ( τ ) = β l ( τ ) (cid:98) g ( τ ) . (2.2)For functions f on R d , and j ∈ N , define the operators L j by (cid:100) L j f ( ξ ) = β j ( | ξ | ) (cid:98) f ( ξ ) , (2.3)and let (cid:101) L j be a modification of L j satisfying (cid:101) L j L j = L j .2.1. V r and related function spaces. It will be convenient to work with theBesov space B /rr, . The Besov spaces B sp,q ( R ) can be defined using the dyadicfrequency decompositions { Λ l } ∞ l =0 on the real line and we have (cid:107) u (cid:107) B sp,q =( (cid:80) ∞ l =0 [2 ls (cid:107) Λ l u (cid:107) p ] q ) /q . From the Plancherel–Polya inequality we know theembedding B /rr, (cid:44) → V r (cid:44) → B /rr, ∞ , (2.4)see [46, Ch.1]. One can also consult the paper by Bergh and Peetre [4] (whohowever work with a different type of variation space when r = 1) or referto [16, Proposition 2.2]. Thus an inequality for the variation operator V Ir A follows if we can control the B /rr, norm of t (cid:55)→ A f ( x, t ).Note that, by our definition, V ( R ) coincides with the space of boundedfunctions of bounded variations. The fundamental theorem of calculus im-plies (cid:107) V E A (cid:107) L p → L q ≤ (cid:107) ∂ t A(cid:107) L p → L q ( L ( E )) , (2.5)so we shall focus on obtaining bounds for the right-hand side when studying V E A . Frequency decomposition in space.
Given j ≥
0, write A t L j f = K j,t ∗ f, (2.6)where L j is as in (2.3), so that (cid:100) K j,t ( ξ ) = (cid:98) σ ( tξ ) β j ( | ξ | ). Note that K j,t is aSchwartz convolution kernel and therefore we restrict our attention to thecase j ≥ Lemma 2.1.
For all N ∈ N , there exists a constant C N > such that | ∂ ςt K j,t ( x ) | (cid:46) ς C N jς j (1 + 2 j (cid:12)(cid:12) | x | − t (cid:12)(cid:12) ) N (2.7) holds for all x ∈ R d , all t > and all ς ∈ N . Consequently, | K j,t ( x ) | (cid:46) N (2 j | x | ) − N if | x | ≥ , t ∈ [1 / , . (2.8)In analogy to the definition of A in (2.1), define A j f ( x, t ) := χ ( t ) A t L j f ( x ) = χ ( t ) K j,t ∗ f ( x ) . We gather some estimates for A j when the inequalities involve L or L ∞ spaces.First, from the trivial fact that (cid:107) A t f (cid:107) ∞ (cid:46) (cid:107) f (cid:107) ∞ uniformly in t ∈ R , oneimmediately has (cid:107)A j f (cid:107) L ∞ ( L ∞ ) (cid:46) (cid:107) f (cid:107) ∞ . (2.9)Moreover, one has the following estimates for L functions. Lemma 2.2.
For ≤ q ≤ ∞ , (cid:107)A j f (cid:107) L q ( L ) + 2 − j (cid:107) ∂ t A j f (cid:107) L q ( L ) (cid:46) (cid:107) f (cid:107) . Proof.
By (2.7) one has (cid:12)(cid:12) A j f ( x, t ) (cid:12)(cid:12) + 2 − j (cid:12)(cid:12) ∂ t A j f ( x, t ) (cid:12)(cid:12) (cid:46) (cid:90) R d | f ( y ) | j (1 + 2 j || x − y | − t | ) N d y (2.10)for all N ∈ N . Integrating in t over the support of χ one sees that, for fixed x , (cid:90) / (cid:12)(cid:12) A j f ( x, t ) (cid:12)(cid:12) d t + 2 − j (cid:90) / (cid:12)(cid:12) ∂ t A j f ( x, t ) (cid:12)(cid:12) d t (cid:46) (cid:90) R d | f ( y ) | (cid:90) / j (1 + 2 j || x − y | − t | ) N d t d y (cid:46) (cid:107) f (cid:107) . This gives the assertion for q = ∞ .For q = 1, the result follows from integrating in x instead, using the decayin (2.10) and taking into account that the integration in t is over [1 / , < q < ∞ follow from combining the above throughYoung’s convolution inequality. (cid:3) ARIATION BOUNDS FOR SPHERICAL AVERAGES 13
Corollary 2.3.
For ≤ r ≤ ∞ , (cid:107)A j f (cid:107) L ∞ ( L r ) (cid:46) j (1 − r ) (cid:107) f (cid:107) . Proof.
Interpolate between (cid:107)A j (cid:107) L ∞ ( L ∞ ) (cid:46) j (cid:107) f (cid:107) , which follows from (2.7), and Lemma 2.2 with q = ∞ . (cid:3) Oscillatory integral representation.
Given m ∈ R , let S m ( R d ) denotethe class of all functions a ∈ C ∞ ( R d ) satisfying | ∂ α a ( ξ ) | (cid:46) α (1 + | ξ | ) m −| α | for all multiindex α ∈ N d and all ξ ∈ R d . Given a ∈ S m ( R ), define T ± j [ a, f ]( x, t ) = (cid:90) R d β j ( | ξ | ) a ( t | ξ | ) e i (cid:104) x,ξ (cid:105)± it | ξ | (cid:98) f ( ξ ) d ξ. (2.11)It is well known that the Fourier transform of the spherical measure is (cid:98) σ ( ξ ) = (2 π ) d/ | ξ | − ( d − / J d − ( | ξ | ) = b ( | ξ | ) + (cid:88) ± b ± ( | ξ | ) e ± i | ξ | , where b ∈ C ∞ c ( R ) is supported in {| ξ | ≤ } and b ± ∈ S − ( d − / ( R ) aresupported in {| ξ | ≥ / } ( c.f. [44, Chapter VIII]). Thus one can write A j f ( x, t ) = 2 − j ( d − / (2 π ) − d (cid:88) ± T ± j [ a ± , f ]( x, t ) χ ( t ) (2.12)where a ± ∈ S ( R ). We note that the kernel estimate (2.7) could also be ob-tained through integration by parts in (2.11) using the above representation.It is clear from the expression of T ± j that ∂ t (cid:0) T ± j [ a, f ]( x, t ) χ ( t ) (cid:1) = T ± j [ a, f ]( x, t ) χ (cid:48) ( t ) + T ± j [ (cid:101) a, f ]( x, t ) χ ( t )where (cid:101) a ( ξ ) = a (cid:48) ( t | ξ | ) | ξ | ± i | ξ | a ( ξ ). This and Plancherel’s theorem yield (cid:107)A j f (cid:107) L ( L ) (cid:46) − j ( d − / (cid:107) f (cid:107) , (cid:107) ∂ t A j f (cid:107) L ( L ) (cid:46) − j ( d − / (cid:107) f (cid:107) . (2.13)2.4. A Stein–Tomas estimate.
In [21], in order to obtain L p bounds for theglobal V r A , the estimate (cid:13)(cid:13)(cid:13)(cid:16) (cid:90) | e it √− ∆ L j f | d t (cid:17) / (cid:13)(cid:13)(cid:13) p (cid:46) j ( d ( − p ) − + ε ) (cid:107) f (cid:107) p (2.14)with ε > d +1) d − ≤ p < ∞ if d ≥
3; it holds for 4 < p < ∞ if d = 2. This statement is closely related to estimates for Stein’s square-function generated by Bochner–Riesz multipliers in [11], [13] and [39], andthe connection is given by the theorem of Kaneko and Sunouchi [22]. Seealso [28] for endpoint bounds and historical remarks, and [27], [26] for recentwork on Stein’s square function. The Stein–Tomas L Fourier restrictiontheorem together with a localization result ( cf . Lemma 4.1 below) yields an analogue of (2.14) with ε = 0 for p ≥ d +1) d − . The method is well known [14]but we include the statement with a proof for completeness. Lemma 2.4.
Let d +1) d − ≤ q ≤ ∞ . Then for all j ≥ , (cid:107)A j f (cid:107) L q ( L ) (cid:46) − jd/q (cid:107) f (cid:107) L . Proof.
We use the oscillatory integral representation in (2.12) and (2.11).We only discuss the estimate for T + j [ a, f ]( x, t ) χ ( t ) and abbreviate it with T j f ( x, t ) (the corresponding estimate for T − j is analogous). It then sufficesto show 2 − j ( d − / (cid:107) T j f (cid:107) L q ( L ) (cid:46) − jd/q (cid:107) f (cid:107) , d +1) d − ≤ q ≤ ∞ . Let (cid:101) T j g ( x, t ) = χ ( t ) (cid:90) R d β j ( | ξ | ) a ( t | ξ | ) e − it | ξ | (cid:98) g ( ξ, t ) e i (cid:104) x,ξ (cid:105) d ξ and observe that in view of the support of χ we have (cid:101) T j g ( · , t ) = 0 for t / ∈ [1 / , g ∈ L p ( L )the inequality (cid:13)(cid:13)(cid:13) (cid:90) (cid:101) T j g ( · , t ) d t (cid:13)(cid:13)(cid:13) (cid:46) j ( dp − d − ) (cid:107) g (cid:107) L p ( L ) , ≤ p ≤ d +1) d +3 (2.15)holds. By Plancherel’s theorem the square of the left-hand side is equal to (cid:90) R d (cid:12)(cid:12)(cid:12) (cid:90) χ ( t ) β j ( | ξ | ) a ( t | ξ | ) e − it | ξ | (cid:98) g ( ξ, t ) d t (cid:12)(cid:12)(cid:12) d ξ = (cid:90) ∞ (cid:90) S d − (cid:12)(cid:12)(cid:12) (cid:90) χ ( t ) β j ( r ) a ( tr ) e − itr (cid:98) g ( rθ, t ) d t (cid:12)(cid:12)(cid:12) d θ r d − d r. We now apply the Stein–Tomas inequality for the Fourier restriction opera-tor for the sphere (valid for 1 ≤ p ≤ d + 1) / ( d + 3)), and see that the lastexpression is dominated by a constant times (cid:90) ∞ (cid:13)(cid:13)(cid:13) (cid:90) χ ( t ) β j ( r ) a ( tr ) e − itr r − d g ( r − · , t ) d t (cid:13)(cid:13)(cid:13) p r d − d r (cid:46) (cid:90) ∞ (cid:13)(cid:13)(cid:13) (cid:90) χ ( t ) β j ( r ) a ( tr ) e − itr g ( · , t ) d t (cid:13)(cid:13)(cid:13) p r dp − d − d r (cid:46) j ( dp − d − ) (cid:13)(cid:13)(cid:13)(cid:16) (cid:90) ∞ (cid:12)(cid:12)(cid:12) (cid:90) χ ( t ) β j ( r ) a ( tr ) e − itr g ( · , t ) d t (cid:12)(cid:12)(cid:12) d r (cid:17) / (cid:13)(cid:13)(cid:13) p (2.16)where in the last inequality we have used Minkowski’s integral inequality.Next, observe that (cid:90) ∞ (cid:12)(cid:12)(cid:12) (cid:90) χ ( t ) β j ( r ) a ( tr ) e − itr g ( x, t ) d t (cid:12)(cid:12)(cid:12) d r = (cid:90) (cid:90) (cid:90) ∞ χ ( t ) χ ( t (cid:48) ) | β j ( r ) | a ( tr ) a ( t (cid:48) r ) e i ( t (cid:48) − t ) r d r g ( x, t ) g ( x, t (cid:48) ) d t d t (cid:48) . ARIATION BOUNDS FOR SPHERICAL AVERAGES 15
We integrate by parts in r and then estimate the absolute value of thedisplayed expression by a constant times (cid:90) (cid:90) j (1 + 2 j | t − t (cid:48) | ) | g ( x, t ) g ( x, t (cid:48) ) | d t d t (cid:48) = (cid:90) ∞−∞ j (1 + 2 j | h | ) (cid:90) | g ( x, t ) g ( x, t + h ) | d t d h (cid:46) (cid:90) | g ( x, t ) | d t. Using this in (2.16) yields (2.15) and hence the assertion. (cid:3)
Frequency decompositions in time.
In order to deduce Besov space es-timates for t (cid:55)→ A j f ( x, t ), we also work with a frequency decomposition inthe t -variable. We extend the definition of Λ l in (2.2) to functions of x and t and apply that decomposition to the operators A j in the t -variable.It is useful to observe that dyadic frequency decompositions in the vari-able dual to t essentially correspond in our situation to dyadic frequencydecompositions in the variables dual to x . To see this, we show that theterms Λ l A j are mostly negligible when | j − l | ≥
10. We writeΛ l A j f ( x, t ) = 2 − j ( d − / (2 π ) − ( d +1) (cid:88) ± (cid:90) R d κ ± j,l ( y, t ) f ( x − y ) d y where, in view of (2.11), one has κ ± j,l ( y, t ) = (cid:90) R (cid:90) R d e i (cid:104) y,ξ (cid:105) + itτ β l ( τ ) β j ( | ξ | ) (cid:90) χ ( s ) a ± ( sξ ) e is ( ±| ξ |− τ ) d s d ξ d τ. (2.17) Lemma 2.5. (i) For every N ∈ N , there exists a finite C N > such that | κ ± j,l ( y, t ) | ≤ C N (1 + | y | + | t | ) − N min { − jN , − lN } , | j − l | ≥ . (2.18) (ii) Suppose ≤ p, r ≤ q ≤ ∞ . Then, there exists a finite C N ( p, q, r ) > such that (cid:107) Λ l A j f (cid:107) L q ( L r ) ≤ C N ( p, q, r ) min { − jN , − lN }(cid:107) f (cid:107) p , | j − l | ≥ . Proof.
Part (i) follows from (2.17) after multiple integration by parts in s and subsequent integration by parts in ξ, τ . Part (ii) is an immediateconsequence of (i) using Minkowski’s and Young’s convolution inequality. (cid:3) The above lemma allows one to only focus on the spatial frequency de-composition when looking for estimates of the type L p → L q ( B /rr, ) for theoperator A in most cases of interest. In particular, we get the following. Corollary 2.6.
Let s ∈ R , ≤ p, q, r ≤ ∞ . Then for all j ∈ N , (cid:107)A j (cid:107) L p → L q ( B sr, ) (cid:46) js (cid:107)A j (cid:107) L p → L q ( L r ) + C N − jN Proof.
We write (cid:107)A j f (cid:13)(cid:13) L q ( B sr, ) ≤ I + II where I = (cid:13)(cid:13)(cid:13) (cid:88) l ≥ | j − l |≤ ls (cid:13)(cid:13)(cid:13) Λ l A j f (cid:13)(cid:13)(cid:13) L r ( R ) (cid:13)(cid:13)(cid:13) L q ( R d ) ,II = (cid:13)(cid:13)(cid:13) (cid:88) l ≥ | j − l | > ls (cid:13)(cid:13)(cid:13) Λ l A j f (cid:13)(cid:13)(cid:13) L r ( R ) (cid:13)(cid:13)(cid:13) L q ( R d ) . Clearly I (cid:46) js (cid:107)A j f (cid:107) L q ( L r ) (cid:46) js (cid:107)A j (cid:107) L p → L q ( L r ) (cid:107) f (cid:107) p and by (ii) in Lemma 2.5 II (cid:46) (cid:88) l ≥ min { − jN , − lN }(cid:107) f (cid:107) p (cid:46) − jN (cid:107) f (cid:107) p . Combining both estimates, the assertion follows. (cid:3)
In certain endpoint estimates in §
6, we use an upgraded version of Corol-lary 2.6 in conjunction with Littlewood–Paley theory, as presented in thenext lemma.
Lemma 2.7.
Let ≤ r < ∞ , ≤ q < ∞ , < p < ∞ such that r, p ≤ q .Let s ∈ R . Assume that for all { f j } j ≥ with f j ∈ L p , (cid:13)(cid:13)(cid:13) (cid:88) j ≥ (cid:107)A j f j (cid:107) L r ( R ) (cid:13)(cid:13)(cid:13) L q ( R d ) (cid:46) (cid:16) (cid:88) j ≥ − jsq (cid:107) f j (cid:107) qp (cid:17) /q (2.19) holds. Then (cid:13)(cid:13) A f (cid:13)(cid:13) L q ( B sr, ) (cid:46) (cid:107) f (cid:107) L p . (2.20) Proof.
Write (cid:107)A f (cid:107) L q ( B sr, ) ≤ I + II , where I and II are as in the proof ofCorollary 2.6 but with an additional sum in the j -parameter. Recall that A j f = A j ( (cid:101) L j f ). Applying the assumption (2.19) in I , one obtains I (cid:46) (cid:13)(cid:13)(cid:13) ∞ (cid:88) j =0 js (cid:107)A j ( (cid:101) L j f ) (cid:107) L r ( R ) (cid:13)(cid:13)(cid:13) L q ( R d ) (cid:46) (cid:16) ∞ (cid:88) j =0 (cid:107) (cid:101) L j f (cid:107) qp (cid:17) q (cid:46) (cid:13)(cid:13)(cid:13)(cid:16) ∞ (cid:88) j =0 | (cid:101) L j f | q (cid:17) q (cid:13)(cid:13)(cid:13) p (cid:46) (cid:13)(cid:13)(cid:13)(cid:16) ∞ (cid:88) j =0 | (cid:101) L j f | (cid:17) (cid:13)(cid:13)(cid:13) p (cid:46) (cid:107) f (cid:107) p since q ≥ < p ≤ q < ∞ ; note that the second line follows fromMinkowski’s inequality, the embedding (cid:96) (cid:44) → (cid:96) q and the Littlewood–Paleyinequality. For the error term II , one applies (ii) in Lemma 2.5 to obtain II (cid:46) N (cid:88) l ≥ (cid:88) j ≥ ls min { − lN , − jN }(cid:107) f (cid:107) p (cid:46) (cid:107) f (cid:107) p for N > s . Combining both estimates, (2.20) follows. (cid:3)
Remark.
The previous lemma also extends to q = ∞ with the obvious no-tational modifications. ARIATION BOUNDS FOR SPHERICAL AVERAGES 17
Bourgain’s interpolation lemma.
For the proof of restricted weak typeinequalities we will repeatedly apply a result of Bourgain [6] that leads torestricted weak type inequalities in certain endpoint situations. We cite theabstract version of this lemma given in [12, § A = ( A , A ), B = ( B , B ) be compatible Banach spaces in the senseof interpolation theory. Let T j : A → B be sublinear operators satisfyingfor all j ∈ Z (cid:107) T j (cid:107) A → B ≤ C jγ , (cid:107) T j (cid:107) A → B ≤ C − jγ , γ , γ > . (2.21)This assumption and real interpolation immediately gives (cid:107) T j (cid:107) A θ,ρ → B θ,ρ = O (1) for all 0 < ρ ≤ ∞ and all θ = γ / ( γ + γ ), but one also gets a weakerconclusion for the sum of the operators. Lemma 2.8.
Suppose (2.21) holds for all j ∈ Z . Then (cid:13)(cid:13)(cid:13) (cid:88) j T j (cid:13)(cid:13)(cid:13) A θ, → B θ, ∞ ≤ C ( γ , γ ) C γ γ γ C γ γ γ .
3. Necessary conditions
In this section we modify known examples for the spherical maximal op-erators to give some necessary conditions for L p → L q boundedness of thelocal variation operator V Ir A . For r > dd − these conditions show that L p → L q boundedness does not hold in the complement of the region P d ( r )in Theorems 1.2 and 1.4 and the complement of Q ( r ) in Theorem 1.6. For1 ≤ r ≤ dd − they show that L p → L q boundedness does not hold in thecomplement of Q d ( r ) defined in Theorem 1.5. They also show that V Ir isunbounded from any L p ( R ) to any L q ( R ) if r <
2, that is, part (iii) in The-orem 1.6. Finally, we also prove sharpness of the sparse bounds in Theorem1.7 up to the endpoints.3.1.
Description of the edges.
It will be helpful to make explicit the equa-tions for the edges of the boundedness regions in the above theorems.(i) Consider the case r > d +1 d ( d − and the region P d ( r ) in Theorem 1.2. Inthis case the point P ( r ) is on the line through (0 ,
0) and Q , which is givenby (cid:8) q = dp (cid:9) . The boundary lines describing P d ( r ) are P ( r ) Q ( r ) = (cid:8) q = dr (cid:9) , Q ( r ) Q = (cid:8) q = p (cid:9) , Q Q = (cid:8) p = d − d (cid:9) ,Q Q = (cid:8) q = d +1 d − p − (cid:9) , Q P ( r ) = (cid:8) q = dp (cid:9) . If d = 2, the points Q and Q coincide, and the lines Q ( r ) Q , Q Q , Q P ( r ) and P ( r ) Q ( r ) describe the quadrangle Q ( r ) in Theorem 1.6, (i).(ii) For the case dd − < r ≤ d +1 d ( d − the point P ( r ) moves to the lineconnecting Q and Q and only the part between P ( r ) and Q will be partof the boundary. Note that for r = d +1 d ( d − the points P ( r ) and Q coincide so that the pentagon degenerates to a quadrangle. As r → dd − the point P ( r ) moves to Q . The boundary lines of P d ( r ) in Theorem 1.4 are givenin this case by Q ( r ) Q = (cid:8) q = p (cid:9) , Q Q = (cid:8) p = d − d (cid:9) ,Q P ( r ) = (cid:8) q = d +1 d − · p − (cid:9) , P ( r ) Q ( r ) = (cid:8) q = p + r ( d − − (cid:9) ,Q ( r ) Q ( r ) = (cid:8) q = dr (cid:9) . It is convenient to note, in view of §
4, that the equation q = p + r ( d − − r = d − (cid:0) q + p (cid:48) (cid:1) .Again, if d = 2, the points Q and Q coincide, and the lines Q ( r ) Q , Q P ( r ), P ( r ) Q ( r ) and Q ( r )( r ) Q ( r ) describe the quadrangle Q ( r ) inTheorem 1.6, (ii).(iii) In the case 1 ≤ r < dd − we now have a quadrangle Q d ( r ) in Theorem1.5, whose boundary lines are Q ( r ) Q ( r ) = (cid:8) p = q (cid:9) , Q ( r ) Q ( r ) = (cid:8) p = 1 − r ( d − (cid:9) ,Q ( r ) Q ( r ) = (cid:8) q = p + r ( d − − (cid:9) , Q ( r ) Q ( r ) = (cid:8) q = dr (cid:9) . We next list our necessary conditions for bounds on V Ir A . We remarkthat the sharpness in the conditions §§ S I .3.2. The condition p ≤ q . This is the standard necessary condition for trans-lation operators mapping L p ( R d ) to L q ( R d ), see [19].3.3. The condition p > dd − . This is (a variant of) Stein’s example for spher-ical maximal functions [43]. Let B be the ball of radius 1 /
10 centered at theorigin and let f ( y ) = B ( y ) | y | − d (log | y | ) − (log log | y | ) − . Then f ∈ L dd − ,q for all q >
1, but for 1 < | x | < t ( x ) = | x | we have A t ( x ) f ( x ) = ∞ . The condition d/q ≥ /p . For the condition d/q ≥ /p we just take thestandard example for the spherical averages [38], namely consider a fixedshell S j, (as in (3.3) below) and g j = S j, so that (cid:107) g j (cid:107) p ≤ − j/p . For | x | ≤ − j − we have A g j ( x ) ≥ c > L q norm over { x : | x | ≤ − j − } we get (cid:107) V Ir Ag j (cid:107) q ≥ − jd/q and obtain the necessity of d/q ≥ /p .3.5. The condition q ≥ d +1( d − p − . This is the standard Knapp example in[38]. Given 0 < δ (cid:28)
1, one tests the maximal operator on f δ being thecharacteristic function of { y : | y (cid:48) | ≤ δ, | y d | ≤ δ } and evaluates A x d f δ ( x ) for | x (cid:48) | ≤ δ and 1 < x d < ARIATION BOUNDS FOR SPHERICAL AVERAGES 19
The condition p ≤ − r ( d − . In view of § r < dd − . For large j define c j,n = − n − j , n = 1 , . . . , N (3.1)where N = 2 j − . Let B j,n be the ball of radius 2 − j − centered at c j,n e d .Let f j ( x ) = (cid:80) Nn =1 ( − n B j,n ( x ), so that (cid:107) f j (cid:107) p (cid:46) N /p − jd/p . Consider R = { ( x (cid:48) , x d ) : | x (cid:48) | ≤ (4 d ) − , ≤ x d ≤ / } . (3.2)Note that for x ∈ R we have | x − c j,n e d | ∈ [1 , | x − c j,n e d | ≥| x d − c j,n | ≥ | x − c j,n e d | ≤ ( | x d − c j,n | + (4 d ) − ) / ≤ x ∈ R pick t n ( x ) = | x − c j,n e d | and observe that there is a constant a > A t ν ( x ) f j ( x ) ≥ a − j ( d − and A t ν − ( x ) f j ( x ) ≤ − a − j ( d − ,and thus | A t ν ( x ) f j ( x ) − A t ν − ( x ) f j ( x ) | ≥ a − j ( d − . Hence, for any r we get V Ir Af ( x ) (cid:38) N /r − j ( d − for x ∈ R and thus forany q > (cid:107) V Ir Af j (cid:107) q (cid:107) f j (cid:107) p (cid:38) N r − p − j ( d − − dp ) Since N = 2 j − the assumption of L p → L q boundedness of V Ir A implies r ≤ d − p (cid:48) or equivalently p ≤ − r ( d − .3.7. The condition q ≥ p + d − r − , i.e. d − ( q + p (cid:48) ) ≥ r . This is a variantof the example in § c j,n be as in (3.1) and P j,n = { y : | y (cid:48) | ≤ − j/ − , | y d − c j,n | ≤ − j − } . Let N ≤ j − . Let f j = (cid:80) Nn =1 ( − n P j,n ( x ).Then (cid:107) f j (cid:107) p (cid:46) N /p − j d +12 p . Let Ω = { x : | x (cid:48) | ≤ − j/ − , ≤ x d ≤ / } so that | Ω | ≈ − j ( d − / . Let t n ( x ) = | x d − c j,n | ∈ [1 , x ∈ Ω, A t ν ( x ) f j ( x ) ≥ a − j ( d − / and A t ν − ( x ) f j ( x ) ≤ − a − j ( d − / for someconstant a >
0. Hence V Ir Af j ( x ) (cid:38) N /r − j d − and thus (cid:107) V Ir Af j (cid:107) q (cid:38) N /r − j d − (1+ q ) . Consequently with N = 2 j − (cid:107) V Ir Af j (cid:107) q (cid:107) f j (cid:107) p (cid:38) N r − p − j d − (1+ q )+ j d +12 p (cid:38) j ( r − d − ( q + p (cid:48) )) . Hence the condition d − ( q + p (cid:48) ) ≥ r is necessary for V Ir A : L p → L q to be bounded. Moreover, as p ≤ q by § L p ( R ) → L q ( R ) bounds hold for r < The condition d/q ≥ /r . Consider the shells S j,n = (cid:8) y : (cid:12)(cid:12) | y | − − n j (cid:12)(cid:12) ≤ − j − (cid:9) . (3.3)We set f j = (cid:80) Nn =1 ( − n S j,n , with N = 2 j − . Then clearly (cid:107) f j (cid:107) p (cid:46) j .For | x | ≤ − j − let t n ( x ) = 1 + n − j ∈ [1 , A t ν ( x ) f j ( x ) ≥ a and A t ν − ( x ) f j ( x ) ≤ − a for some a independent of j . Hence V Ir f ( x ) (cid:38) N /r ≈ j/r for | x | ≤ − j and thus (cid:107) V Ir f j (cid:107) q (cid:38) j ( r − dq ) . This implies the necessityof the condition 1 /r ≤ d/q . Remark.
An alternative (more complicated) example for the condition d/q ≥ /r is in [21, § Sharpness of the sparse bounds.
The sparse domination result in The-orem 1.7 is sharp, and this is immediate from the examples just describedin this section. The argument, shown by Lacey in [23, Section 5] for thespherical maximal function, can be extended in our context and even moregeneral ones [2, Proposition 7.2].We exemplify this considering the example in § N =2 j − . With f j as in this example we have | f j | = U where U is the union ofthe balls B j,n which is essentially a 2 − j -neighborhood of the x d -axis segment[ − / , V r Af j is evaluated at R as in (3.2). Then for large j we have (cid:104) V r Af j , R (cid:105) = (cid:90) R d V r Af ( x ) R ( x ) d x (cid:38) j ( r − d +1) . On the other hand, suppose that p < q and the sparse bound (cid:90) R d V r Af j ( x ) R ( x ) d x ≤ C sup S :sparse Λ S p,q (cid:48) ( f, R )holds for some positive C , with Λ S p,q (cid:48) ( f, g ) = (cid:80) Q ∈ S | Q |(cid:104) f j (cid:105) Q,p (cid:104) R (cid:105) Q,q (cid:48) . Bythe definition of supremum there is a sparse collection S such that (cid:90) R d V r Af j ( x ) R ( x ) d x ≤ C (cid:88) Q ∈ S | Q |(cid:104) f j (cid:105) Q,p (cid:104) R (cid:105) Q,q (cid:48) . It is crucial in the example thatdist(supp( f j ) , R ) ≥ l ≥ O (1) cubes of sidelength 2 l contributing. For each such term we can estimate | Q |(cid:104) f j (cid:105) Q,p (cid:104) R (cid:105) Q,q (cid:48) (cid:46) | Q | q − p − j d − p and by summing over all terms (taking advantage of p < q ) we obtain2 j ( r − d +1) (cid:46) (cid:104) V r Af j , R (cid:105) = (cid:90) R d V r Af j ( x ) R ( x ) d x (cid:46) C − j d − p ARIATION BOUNDS FOR SPHERICAL AVERAGES 21 and letting j → ∞ we obtain the same necessary condition as in § p ≤ − r ( d − .The remaining examples in §§ L p → L q ( L r ) estimates for A j In this section we prove L p → L q ( L r ) bounds for the dyadic frequencylocalized operators A j in the closure of the regions P d ( r ) and Q d ( r ) fea-turing in Theorems 1.2, 1.4, 1.5 and 1.6. This will lead to the proofs for L p → L q bounds for V Ir A if ( p , q ) belongs to the interior of P d ( r ) and Q d ( r ) respectively, as well as several restricted weak-type results throughBourgain’s interpolation trick.4.1. Localization.
The following observation relies on the localization prop-erty (2.6) of the kernel K j,t . Lemma 4.1. (i) For p ≤ p ≤ q ≤ q , ≤ r ≤ ∞ , and every N ∈ N , (cid:107)A j (cid:107) L p → L q ( L r ) (cid:46) (cid:107)A j (cid:107) L p → L q ( L r ) + C N − jN . (ii) For r ≤ r , ≤ p ≤ q ≤ ∞ , (cid:107)A j (cid:107) L p → L q ( L r ) (cid:46) (cid:107)A j (cid:107) L p → L q ( L r ) . Proof.
Assume that (cid:107)A j (cid:107) L p → L q ( L r ) < ∞ . Let f ∈ L p . For z ∈ Z d let Q z = (cid:81) di =1 [ z i , z i + 1). Let Q ∗ z be a cube centered at z with side-length 20 d .Write f = (cid:80) z f z with f z = f Q z and estimate (cid:107)A j f (cid:107) L q ( L r ) ≤ (cid:13)(cid:13)(cid:13) (cid:88) z Q ∗ z A j f z (cid:13)(cid:13)(cid:13) L q ( L r ) + (cid:13)(cid:13)(cid:13) (cid:88) z R d \ Q ∗ z A j f z (cid:13)(cid:13)(cid:13) L q ( L r ) = I + II.
Since the Q ∗ z have bounded overlap, by H¨older’s inequality for q ≤ q , I (cid:46) (cid:16) (cid:88) z (cid:107) Q ∗ z A j f z (cid:107) q L q ( L r ) (cid:17) /q (cid:46) (cid:16) (cid:88) z (cid:107)A j f z (cid:107) q L q ( L r ) (cid:17) /q . Applying the bound for the operator A j , (cid:16) (cid:88) z (cid:107)A j f z (cid:107) q L q ( L r ) (cid:17) /q (cid:46) (cid:107)A j (cid:107) L p → L q ( L r ) (cid:16) (cid:88) z (cid:107) f z (cid:107) q L p (cid:17) /q and, since p ≤ p ≤ q , we also have (cid:16) (cid:88) z (cid:107) f z (cid:107) q L p (cid:17) /q (cid:46) (cid:16) (cid:88) z (cid:107) f z (cid:107) q L p (cid:17) /q (cid:46) (cid:16) (cid:88) z (cid:107) f z (cid:107) p L p (cid:17) /p (cid:46) (cid:107) f (cid:107) p . Moreover, by (2.8) with
N > d , II ≤ (cid:16) (cid:90) (cid:104) (cid:90) | y − x |≥ (2 j | x − y | ) − N | f ( y ) | d y (cid:105) q d x (cid:17) /q (cid:46) N − jN (cid:107) f (cid:107) p . p q q = p p q ρ min ( q )1/ ρ max ( q ) q Figure 9.
Interpolation and localization lemmas. If (cid:107)A j (cid:107) L p → L q ( L p ) (cid:46) − jd/q , then (cid:107)A j (cid:107) L p → L q ( L p ) (cid:46) − jd/q in the blue triangle and (cid:107)A j (cid:107) L p → L q ( L ρ max( q ) ) (cid:46) − jd/q in thered triangle.Combining the two estimates we obtain (cid:107)A j f (cid:107) L q ( L r ) (cid:46) (cid:0) (cid:107)A j (cid:107) L p → L q ( L r ) + C N − jN (cid:1) (cid:107) f (cid:107) p , which is the assertion in part (i).Part (ii) is immediate and simply follows from H¨older’s inequality in the t -variable. (cid:3) Interpolation.
Lemma 2.4 can be extended to a larger range of expo-nents by interpolation with (2.9) and Lemma 2.2 and by the localizationproperty in Lemma 4.1. We state this in more generality; see Figure 9.
Lemma 4.2.
Let p and q such that ≤ p ≤ q ≤ ∞ . Assume that sup j ≥ jd/q (cid:107)A j (cid:107) L p → L q ( L p ) ≤ C < ∞ . (4.1) Let q ≤ q ≤ ∞ and define ρ min ( q ) and ρ max ( q ) by − ρ min ( q ) = q q (cid:16) − p (cid:17) , ρ max ( q ) = q q p . (4.2) Assume that ρ min ( q ) ≤ p ≤ q and < r ≤ min { p, ρ max ( q ) } . Then sup j ≥ jd/q (cid:107)A j (cid:107) L p → L q ( L r ) < ∞ . Proof.
Note that ρ min ( q ) ≤ ρ max ( q ) when q ≥ q , with strict inequality when q > q , and ρ min ( q ) = ρ max ( q ) = p . Assume q > q and let ϑ = 1 − q /q .Note that (1 − ϑ ) /p = 1 /ρ max ( q ) and (1 − ϑ ) /p + ϑ = 1 /ρ min ( q ). Weinterpolate (4.1) with the inequalitysup j ≥ (cid:107)A j (cid:107) L p → L ∞ ( L p ) < ∞ , ≤ p ≤ ∞ ARIATION BOUNDS FOR SPHERICAL AVERAGES 231 p q d − d +1) AB B CD E d − d Figure 10.
Regions for L p → L q ( L r ) bounds for the singlescale A j for 0 < r ≤
1. As r increases the regions shrink dueto the constraints r ≤ p or r ≤ q ( d − d +1 .for the choices p = 1 and p = ∞ (by Lemma 2.2 and (2.9)) and obtainthe L p → L q ( L p ) inequality for p = ρ min ( q ) and p = ρ max ( q ). A furtherinterpolation givessup j ≥ (cid:107)A j (cid:107) L p → L q ( L p ) (cid:46) (cid:0) j ≥ (cid:107)A j (cid:107) L p → L q ( L p ) (cid:1) , ρ min ( q ) ≤ p ≤ ρ max ( q ) . We now combine this with Lemma 4.1 and see that the L p → L q ( L r ) es-timates hold when ρ min ( q ) ≤ p ≤ ρ max ( q ) and r ≤ p and moreover when ρ min ( q ) ≤ r ≤ ρ max( q ) and r ≤ p ≤ q . (cid:3) Bounds for A j . The previous lemma and the estimates in § Proposition 4.3.
Let d ≥ .(A) Let ≤ p ≤ , p ≤ q ≤ p (cid:48) and < r ≤ p . Then (cid:107)A j f (cid:107) L q ( L r ) (cid:46) − j ( d − /p (cid:48) (cid:107) f (cid:107) L p . (B) Let ≤ p ≤ q ≤ d +1) d − . Let < r ≤ . Then (cid:107)A j f (cid:107) L q ( L r ) (cid:46) − j d − ( q + ) (cid:107) f (cid:107) L p . (C) Let ≤ p ≤ , d − d +1 1 p (cid:48) ≤ q ≤ p (cid:48) and < r ≤ p. Then (cid:107)A j f (cid:107) L q ( L r ) (cid:46) − j d − ( q + p (cid:48) ) (cid:107) f (cid:107) L p . (D) Let d +1) d − ≤ q ≤ ∞ , d − d +1 1 p ≤ q ≤ p and < r ≤ q ( d − d +1 . Then (cid:107)A j f (cid:107) L q ( L r ) (cid:46) − jd/q (cid:107) f (cid:107) L p . (E) Let d +1) d − ≤ q ≤ ∞ , q ≤ d − d +1 1 p , q ≤ p ≤ − d +1 d − q , and < r ≤ p .Then (cid:107)A j f (cid:107) L q ( L r ) (cid:46) − jd/q (cid:107) f (cid:107) L p . Proof.
The bounds in (A) for r = p follow from interpolation of Lemma 2.2and the L -estimate (2.13), whilst the remaining values of 0 < r < p followfrom (ii) in Lemma 4.1.The bounds in (D) and (E) are an application of Lemma 4.2 with p = 2, q = d +1) d − , which is the estimate in Lemma 2.4.The bounds in (C) follow from interpolation of those in (A) if q = p (cid:48) andthose in (E) if q = p (cid:48) d − d +1 , 1 ≤ p ≤ L estimate(2.13) with the L p → L p ( L ) estimate in (D) for p = d +1) d − , and a furtherinterpolation of those with the estimates in (C) for p = 2. (cid:3) The above bounds on (A), (C) and (E) are sharp. However, the boundsin (B) and the r -range in (D) can be improved; for example, if informationon the local smoothing phenomenon for the wave equation is known. Recallthat these estimates, first noted by Sogge in [41], are of the type (cid:13)(cid:13)(cid:13)(cid:16) (cid:90) | e it √− ∆ L j f | p d t (cid:17) /p (cid:13)(cid:13)(cid:13) L p (cid:46) j (¯ s p − σ ) (cid:107) f (cid:107) L p (4.3)for some σ > < p < ∞ , where ¯ s p := ( d − (cid:0) − p (cid:1) . It is conjecturedthat (4.3) holds for all σ < σ p , where σ p := (cid:40) /p if dd − ≤ p < ∞ , ¯ s p if 2 ≤ p ≤ dd − . This conjecture is strongest at p = dd − . After contributions by many, ithas recently been solved by Guth, Wang and Zhang [17] for d = 2, and isknown to hold for all p ≥ d +1) d − if d ≥ σ = 1 /p should hold if p > d/ ( d − d ≥
4. The validity of the local smoothing conjecturewould imply the following bounds on spherical averages on the region (B).We remark that these improved bounds are only relevant for our variationalbounds if d = 2 ,
3; for d ≥ Proposition 4.4.
Let d ≥ . Assume that the local smoothing conjectureholds, that is, (4.3) holds at p = dd − for all σ < /p. ARIATION BOUNDS FOR SPHERICAL AVERAGES 25 ( B ) If d − d +1 1 p (cid:48) ≤ q ≤ p and < q ≤ dd − , < p ≤ dd − and < r ≤ p ,then (cid:107)A j f (cid:107) L q ( L r ) (cid:46) − j d − ( q + p (cid:48) )+ jε (cid:107) f (cid:107) L p for all ε > .( B ) If q ≤ min { d − d +1 1 p (cid:48) , p } and dd − ≤ q ≤ d +1) d − and < r ≤ p , then (cid:107)A j f (cid:107) L q ( L r ) (cid:46) − jd/q + jε (cid:107) f (cid:107) L p for all ε > .In particular, the above estimates hold for d = 2 .Proof. By the oscillatory integral representation in (2.12) and (2.11), theestimate (4.3) implies (cid:107)A j f (cid:107) L p ( L p ) (cid:46) − j d − + jε (cid:107) f (cid:107) L p (4.4)for p = dd − . Interpolation of (4.4) and Lemma 2.4 yields (cid:107)A j f (cid:107) L q ( L p ) (cid:46) − jd/q + jε (cid:107) f (cid:107) L p (4.5)for q = d − d +1 1 p (cid:48) and 2 < p ≤ dd − ≤ q < d +1) d − . Moreover, interpolation of(4.4) and the L -estimate (2.13) yields (cid:107)A j (cid:107) L p ( L p ) (cid:46) − j d − + jε (cid:107) f (cid:107) L p (4.6)for 2 < p ≤ dd − . The region (B ) then follows from interpolating (4.5) and(4.6).For the region (B ), interpolate (4.4) and (2.9) to obtain (cid:107)A j f (cid:107) L q ( L q ) (cid:46) − jd/q + jε (cid:107) f (cid:107) L q (4.7)for all dd − ≤ q ≤ ∞ . A further interpolation of (4.7) with (4.5) for dd − ≤ q ≤ d +1) d − yields the estimates in (B ).The assertion for d = 2 follows since the local smoothing assumption wasestablished in [17]. (cid:3) The range of r in the estimates in (D) can also be improved to 0 < r ≤ p using the known local smoothing estimates at p = d +1) d − for all σ < /p . Forour variational problem, this only becomes relevant if d = 2, as otherwisethe results in Proposition 4.3 will suffice. We note that the use of such localsmoothing estimates induces an ε -loss with respect to (D) in Proposition4.3, although this will have no consequences on our proof in d = 2. The ε -loss in the forthcoming proposition can be removed if p > d − d − when d ≥ Proposition 4.5 (Improved bounds in (D)) . Let d ≥ . Let d +1) d − ≤ q ≤∞ , d − d +1 1 p ≤ q ≤ p and r ≤ p . Then (cid:107)A j f (cid:107) L q ( L r ) (cid:46) − j ( d/q − ε ) (cid:107) f (cid:107) p for all ε > .Proof. By (2.12), the estimates (4.3) for p ≥ d +1) d − imply that, given any ε > (cid:107)A j f (cid:107) L p ( L p ) (cid:46) − j ( d/q − ε ) (cid:107) f (cid:107) L q holds for all d +1) d − ≤ q ≤ ∞ . It then suffices to interpolate this with theestimates in Proposition 4.3, (D), when q = p ( d +1) d − and r = p . (cid:3) Bounds for V Ir A j . Let 1 ≤ r ≤ ∞ . By the embedding (2.4) and Corol-lary 2.6, V r A maps L p ( R d ) to L q ( R d ) if there exists an ε > (cid:107)A j f (cid:107) L q ( L r ) (cid:46) − j ( r + ε ) (cid:107) f (cid:107) L p , (4.8)for all f ∈ L p . This will suffice to show all the bounds in the interiors of P d ( r ) , Q d ( r ) claimed in Theorems 1.2, 1.4, 1.5 and 1.6.We start with the case d ≥
3. We will only have to identify in each region A − E of Proposition 4.3 the conditions under which (4.8) holds and to relatethis to the corresponding statements in the theorems in the introduction. Proposition 4.6.
Let d ≥ . The inequality (4.8) holds for some ε > under the following conditions on ≤ p, q ≤ ∞ , < r ≤ ∞ :(A’) ≤ p ≤ , p ≤ q ≤ p (cid:48) , and ◦ dd − < r ≤ p ; or ◦ d − < r ≤ dd − and p < − d − r .(B’) ≤ p ≤ q ≤ d +1) d − and ◦ d +1) d ( d − < r ≤ ; or ◦ d − < r ≤ d +1) d ( d − and q > d − r − .(C’) ≤ p ≤ , d − d +1 1 p (cid:48) ≤ q ≤ p (cid:48) , and ◦ d +1 d ( d − < r ≤ p and q > d +1 d − p − , p < d − d ; or ◦ dd − < r ≤ min { d +1 d ( d − , p } and q > d +1 d − p − , q > p + r ( d − − ;or ◦ d − < r ≤ dd − and q > p + r ( d − − .(D’) d +1) d − ≤ q ≤ ∞ , d − d +1 1 p ≤ q ≤ p and q > dr for d +1) d ( d − < r ≤ q ( d − d +1 .(E’) d +1) d − ≤ q ≤ ∞ , q ≤ d − d +1 1 p , q ≤ p ≤ − d +1 d − q and q > dr for d +1) d ( d − < r ≤ p .Proof. It suffices to check that the exponents appearing in the inequalities A − E in Proposition 4.3 are strictly greater than 1 /r under the claimedconditions.(A’) The exponent in (A), Proposition 4.3, is d − p (cid:48) . Note that d − p (cid:48) > r issatisfied if dd − < r ≤ p . Moreover, it also holds if p < ( d − r − d − r and ARIATION BOUNDS FOR SPHERICAL AVERAGES 27 r ≤ dd − . The additional constraint r > d − follows since p ≥ d ≥ d − ( q + ). Note that d − ( q + ) > r is satisfied if d +1) d ( d − < r ≤
2, as q ≤ d +1) d − . Moreover, it alsoholds if q > d − r − and r ≤ d +1) d ( d − . The additional constraint r > d − follows since q ≥ d ≥ d − ( q + p (cid:48) ). Note that d − ( q + p (cid:48) ) > r is satisfied if d +1 d ( d − < r ≤ p , as q ≥ d − d +1 1 p (cid:48) . The additionalconstraint q > d +1 d − p − r ≤ p . Note that this and q ≥ p (cid:48) ,also yield the additional constraint p < d − d .For the remaining values r ≤ d +1 d ( d − , it simply holds by the assump-tion q > p + r ( d − −
1. Note that r ≤ p is automatically satisfiedif r ≤ dd − . The lower bound r > d − follows from the assumption q > p + r ( d − − q ≥ p (cid:48) and p ≤
2. This yields d − < r ≤ p ≤ d ≥ dq . Note that dq > r is triviallysatisfied if q > dr . The lower bound r > d +1) d ( d − , follows from q ≤ d +1) d − . Note that when combined with r ≤ q ( d − d +1 requires d ≥ dq . Note that dq > r is triviallysatisfied if q > dr . The constraint r > d +1) d ( d − follows from q ≥ d +1) d − .Note the above constraints combined yield the additional condition d ≤ rq ≤ d − d +1 , which requires d ≥ (cid:3) We next turn to the case d = 2. As observed in the proof of the previousproposition, the bounds in Proposition 4.3 do not yield any bound of thetype (4.8) for d = 2. We use instead the upgraded bounds from Propositions4.4 and 4.5. Proposition 4.7.
Let d = 2 . The inequality (4.8) holds for some ε > under the following conditions on ≤ p, q ≤ ∞ , < r ≤ ∞ :(B ’) p (cid:48) ≤ q ≤ p and < q ≤ , < p ≤ , and ◦ / < r ≤ p ; or ◦ < r ≤ min { / , p } and q > p + r − .(B ’) q ≤ min { p (cid:48) , p } , ≤ q ≤ and q > r for < r ≤ p .(D’) ≤ q ≤ ∞ , p ≤ q ≤ p and q > r for < r ≤ p .Proof. As in Proposition 4.6, it suffices to check that the exponents appear-ing in the inequalities B , B in Proposition 4.4 and in Proposition 4.5 arestrictly greater than 1 /r under the claimed conditions. (B ’) The exponent in (B ’), Proposition 4.4 is ( q + p (cid:48) ) − ε . Choosing ε > ( q + p (cid:48) ) − ε > r is satisfied using q ≤ p (cid:48) and 5 / < r ≤ p . If r ≤ min { / , p } , the required condition followssimply by assumption choosing ε > r > ( q + p (cid:48) ) > r and p ≤ q .(B ’) The exponent in (B ’), Proposition 4.4 is 2 /q − ε . Choosing ε > /q − ε > r is trivially satisfied by the assumption q > r . The lower bound r > q > r and q ≥ /q − ε . Choosing ε > /q − ε > r is trivially satisfied by the assumption q > r . Note that the lower bound r > q > r and q ≥ (cid:3) Combining Propositions 4.6 and 4.7 with the observations in § V Ir A for all r ≥
1. We use the trivial fact that L q ( V r ) is embedded in L q ( V r ) for r < r , which allows to overcome the r ≤ p or r ≤ q ( d − d +1 constraints in the above Propositions. Corollary 4.8.
Let d ≥ . V Ir A : L p → L q is bounded if one of the followingconditions is satisfied:(i) ( p , q ) belongs to the open line segment ( Q ( r ) , Q ) or the interior ofthe domain P d ( r ) in Theorem 1.2 ( r > d +1 d ( d − ).(ii) ( p , q ) belongs to the open line segment ( Q ( r ) , Q ) or the interior ofthe domain P d ( r ) in Theorem 1.4 ( dd − < r ≤ d +1 d ( d − ).(iii) ( p , q ) belongs to the open line segment ( Q ( r ) , Q ( r )) or the interiorof the domain Q d ( r ) in Theorem 1.5 ( ≤ r ≤ dd − for d ≥ or < r ≤ for d = 3 ). Corollary 4.9.
Let d = 2 . V Ir A : L p → L q is bounded if one of the followingconditions is satisfied:(i) ( p , q ) belongs to the open line segment ( Q ( r ) , Q ) or the interior ofthe domain Q ( r ) in Theorem 1.6, (i) ( r > ).(ii) ( p , q ) belongs to the open line segment ( Q ( r ) , Q ) or the interior ofthe domain Q ( r ) in Theorem 1.6, (ii) ( < r ≤ ). Various endpoint bounds.
We shall discuss various endpoint boundsthat can be obtained by interpolation (in particular Bourgain’s interpolationlemma as formulated in § Q . Lemma 4.10.
Let d ≥ , r > dd − . Let p = dd − , q = d . ARIATION BOUNDS FOR SPHERICAL AVERAGES 29
Then A : L p , → L q , ∞ ( B /rr, ) is bounded. Consequently, V Ir A is ofrestricted weak type at Q in Theorems 1.2 and 1.4.Proof. By standard embedding theorems, we can assume r ≤
2. For r > dd − we have d − r (cid:48) − r > . We have from Corollary 2.3 and Proposition 4.3, (A), (cid:107)A j f (cid:107) L ∞ ( L r ) (cid:46) j (1 − /r ) (cid:107) f (cid:107) , (cid:107)A j f (cid:107) L r (cid:48) ( L r ) (cid:46) − j d − r (cid:48) (cid:107) f (cid:107) r , and by Corollary 2.6 (cid:107)A j f (cid:107) L ∞ ( B /rr, ) (cid:46) j (cid:107) f (cid:107) , (cid:107)A j f (cid:107) L r (cid:48) ( B /rr, ) (cid:46) − j ( d − r (cid:48) − r ) (cid:107) f (cid:107) r . The lemma then follows by applying § V r A is a simple corollary in view of (2.4). (cid:3) A similar argument yields a restricted weak type bound at Q . Lemma 4.11.
Let d ≥ , r > d +1 d ( d − and p = d +1 d ( d − , q = d +1 d − .Then A : L p , → L q , ∞ ( B /rr, ) is bounded. Consequently, V Ir A is ofrestricted weak type at Q in Theorem 1.2.Proof. By standard embedding theorems, we can assume r ≤
2. By assump-tion on r we have d ( d − d +1) r (cid:48) − r >
0. It then suffices to interpolate using § (cid:107)A j f (cid:107) L ∞ ( B /rr, ) (cid:46) j (cid:107) f (cid:107) L (cid:107)A j f (cid:107) L q ◦ ( B /rr, ) (cid:46) − j ( dq ◦ − r ) (cid:46) (cid:107) f (cid:107) L p ◦ with p ◦ = r, q ◦ = d +1 d − r (cid:48) ;the last inequality follows from Proposition 4.3, (E). (cid:3) Corollary 4.12.
Let d ≥ . Then the following hold:(i) V Ir A : L p → L q is bounded if (1 /p, /q ) belongs to the open segment ( Q , Q ) in Theorem 1.2 ( r > d +1 d ( d − ).(ii) V Ir A : L p, → L q is bounded if (1 /p, /q ) belongs to the half-opensegment [ Q , Q ) in Theorems 1.2 and 1.4 ( r > dd − ).Proof. Part (i) just follows from interpolation between Lemma 4.10 and 4.11.For part (ii), let p = dd − and fix q = dd − and q = d . For z ∈ Z d , let Q z = (cid:81) di =1 [ z i , z i + 1) and let Q ∗ z be a cube centered at z with sidelength 20 d .Write f = (cid:80) z f z with f z = f Q z . As V Ir A is local and the Q ∗ z have boundedoverlap, by H¨older’s inequality (cid:107) V Ir Af (cid:107) L q , ∞ ≤ (cid:16) (cid:88) z (cid:107) Q ∗ z V Ir Af z (cid:107) q L q , ∞ (cid:17) /q ≤ (cid:16) (cid:88) z (cid:107) Q ∗ z V Ir Af z (cid:107) q L q , ∞ (cid:17) /q . By Lemma 4.10, the right-hand side is further bounded by (cid:16) (cid:88) z (cid:107) f z (cid:107) q L p, (cid:17) /q (cid:46) (cid:107) f (cid:107) p, , as p = q = p = dd − . This implies that V Ir A is of restricted weak typeat Q if r > dd − . By interpolation between Q and Q , one has that V Ir A is of restricted strong type on the open line segment ( Q , Q ). Finally, therestricted strong type at Q follows from the above localization argument,but using any of the just obtained L p, → L q inequalities for q < q < q instead of the L p, → L q , ∞ . (cid:3) Remark.
One can obtain that V Ir A is of restricted weak type at Q in Theo-rems 1.2 and 1.4 ( r > d/ ( d − § (cid:107)A j f (cid:107) L ( B /rr, ) (cid:46) j (cid:107) f (cid:107) (cid:107)A j f (cid:107) L ( B /rr, ) (cid:46) − j ( d − / j/r (cid:107) f (cid:107) . Interpolation with the restricted weak type bound at Q yields the restrictedstrong type bounds on the open line segment ( Q , Q ). However, in orderto deduce the restricted strong type at Q we need to argue with the local-ization argument presented in the proof of Corollary 4.12 above.We next address the claimed bounds for V I A in Theorem 1.5. Lemma 4.13.
Let d ≥ . The operator ∂ t A maps L d − d − , boundedly to L d − , ∞ ( L ) . Consequently, V Ir A is of restricted weak type at Q (1) in The-orem 1.5.Proof. We have (cid:107) ∂ t A j f (cid:107) L ( L ) (cid:46) (cid:107) ∂ t A j f (cid:107) L ( L ) (cid:46) − j d − (cid:107) f (cid:107) . We interpo-late the estimates (obtained from Corollary 2.3 and Proposition 4.3 togetherwith Corollary 2.6) (cid:107) ∂ t A j f (cid:107) L ∞ ( L ) (cid:46) j (cid:107) f (cid:107) (cid:107) ∂ t A j f (cid:107) L ( L ) (cid:46) − j d − (cid:107) f (cid:107) and obtain the conclusion by application of § (cid:3) Corollary 4.14.
Let d ≥ . The operator V Ir A : L p, → L q is bounded if (1 /p, /q ) belongs to the half-open line segment [ Q (1) , Q (1)) in Theorem1.5.Proof. The restricted strong type bounds on [ Q (1) , Q (1)) can be obtainedas in Corollary 4.12. (cid:3) Lemma 4.15.
Let d = 3 . The operator V I A is bounded on L ( R ) .Proof. By (2.5) we have (cid:107) V I A (cid:107) ≤ (cid:13)(cid:13)(cid:13) (cid:90) I | ∂ t A f ( · , t ) | d t (cid:13)(cid:13)(cid:13) (cid:46) (cid:16) (cid:90) (cid:90) | ∂ t A t f | d x d t (cid:17) / (cid:46) (cid:107) f (cid:107) , by (2.13) and orthogonality. (cid:3) ARIATION BOUNDS FOR SPHERICAL AVERAGES 31
5. A maximal operator
We first introduce an auxiliary maximal function which will be crucial inthe proof of the endpoint bounds in § L ∈ Z let Q L be the family of all cubes in R d with side length in(2 L − , L ]. Given Q we write L ( Q ) = L if Q ∈ Q L . (5.1)We use the slashed integral to denote an average, i.e. \ (cid:90) Q g ( y ) d y = 1 | Q | (cid:90) Q g ( y ) d y. For x ∈ R d we let Q L ( x ) be the collection of all Q ∈ Q L containing x . Given n = 0 , , , . . . and a sequence of functions F = { f j } j ≥ , define the maximalfunction M r,n F ( x ) = sup j ≥ n sup Q ∈Q n − j ( x ) \ (cid:90) Q (cid:16) (cid:90) |A j f j ( y, t ) | r d t (cid:17) /r d y. (5.2)The following result should be compared with Lemma 4.2. Away fromthe right boundary of the region in that lemma, we gain a crucial factor of2 − nε . Related statements can be found in [36], [34] (see also [18] for dualversions). Proposition 5.1.
Let p and q such that < p ≤ q < ∞ . Assume that sup j ≥ jd/q (cid:107)A j (cid:107) L p → L q ( L p ) < ∞ . (5.3) Let q < q ≤ ∞ and ρ max ( q ) = q q p and ρ min ( q ) = 1 − q q (1 − p ) . Assumethat ρ min ( q ) < p ≤ q and (cid:40) r ≤ p if ρ min ( q ) < p < ρ max ( q ) ,r < ρ max ( q ) if ρ max ( q ) ≤ p ≤ q. (5.4) Then there exists ε ( p, q, r ) > such that (cid:107) M r,n F (cid:107) q ≤ C p,q,r − nε ( p,q,r ) (cid:16) (cid:88) j ≥ n − jd (cid:107) f j (cid:107) qp (cid:17) /q . (5.5)For the proof we first observe a uniform estimate in n . Lemma 5.2.
Let p ≤ q and assume (5.3) holds. Let q > and q ≤ q ≤ ∞ . Let ρ min ( q ) , ρ max ( q ) be as in (4.2) , and let ρ min ( q ) ≤ p ≤ q and < r ≤ min { p, ρ max ( q ) } . Then (cid:107) M r,n F (cid:107) q (cid:46) (cid:16) (cid:88) j ≥ n − jd (cid:107) f j (cid:107) qp (cid:17) /q . (5.6) Proof.
Let M HL denote the Hardy–Littlewood maximal function. Then | M r,n F ( x ) | (cid:46) sup x ∈ Q \ (cid:90) Q sup j ≥ n (cid:16) (cid:90) |A j f j ( y, t ) | r d t (cid:17) /r d y ≤ M HL (cid:2) sup j ≥ n (cid:107)A j f j (cid:107) L r ( R ) (cid:3) ( x )and therefore, since r ≤ q and q > (cid:107) M r,n F (cid:107) q (cid:46) (cid:13)(cid:13) sup j ≥ n (cid:107)A j f j (cid:107) L r ( R ) (cid:13)(cid:13) q (cid:46) (cid:13)(cid:13)(cid:13)(cid:16) (cid:88) j ≥ n (cid:107)A j f j (cid:107) qL r ( R ) (cid:17) /q (cid:13)(cid:13)(cid:13) q = (cid:16) (cid:88) j ≥ n (cid:107)A j f j (cid:107) qL q ( L r ) (cid:17) /q (cid:46) (cid:16) (cid:88) j ≥ n − jd (cid:107) f j (cid:107) qp (cid:17) /q ;here in the last step we have used Lemma 4.2. (cid:3) We now show how to gain over this inequality in the special case r = p . Lemma 5.3.
Let p ≤ q and assume (5.3) holds. Then for q ≤ q ≤ ∞ , p ≤ p ≤ q (cid:107) M p ,n F (cid:107) q (cid:46) − nd ( q − q ) (cid:16) (cid:88) j ≥ n − jd (cid:107) f j (cid:107) qp (cid:17) /q . (5.7) Proof.
We use real interpolation for the sublinear operator M p ,n . Then(5.7) follows from (cid:107) M p ,n F (cid:107) q (cid:46) (cid:16) (cid:88) j ≥ n − jd (cid:107) f j (cid:107) q p (cid:17) /q , p ≤ p ≤ q , (5.8a)and (cid:107) M p ,n F (cid:107) ∞ (cid:46) − nd/q sup j ≥ n (cid:107) f j (cid:107) p , p ≤ p ≤ ∞ . (5.8b)Note that (5.8a) immediately follows by Lemma 5.2.We now show (5.8b). Fix x ∈ R d , j ≥ n and Q ∈ Q n − j ( x ). Let R x be acube of diameter 20 d centered at x . Then split \ (cid:90) Q (cid:16) (cid:90) |A j f j ( y, t ) | p d t (cid:17) /p d y ≤ I ( x ) + II ( x )where I ( x ) = \ (cid:90) Q (cid:16) (cid:90) |A j [ R x f j ]( y, t ) | p d t (cid:17) /p d y,II ( x ) = \ (cid:90) Q (cid:16) (cid:90) |A j [ R (cid:123) x f j ]( y, t ) | p d t (cid:17) /p d y. ARIATION BOUNDS FOR SPHERICAL AVERAGES 331 p q q = p p q ρ min ( q )1/ ρ max ( q ) q Figure 11.
Bounds for M r,n . At r = p = p , q < q ≤ ∞ ,we have a gain in n given by Lemma 5.3. Interpolation withthe uniform estimates from Lemma 5.2 for r = p = ρ min ( p )and r = p = ρ max ( q ) yields the estimates in the interior ofthe blue triangle. The bounds in the red triangle follow bythe localization argument.Using H¨older’s inequality, then the assumption (5.3) and then again H¨older’sinequality we get I ( x ) ≤ (cid:16) | Q | (cid:90) (cid:107)A j [ R x f j ]( y, · ) (cid:107) q p d y (cid:17) /q (cid:46) | Q | − /q − jd/q (cid:107) R x f j (cid:107) p (cid:46) − nd/q (cid:107) R x f j (cid:107) p (cid:46) − nd/q (cid:107) f j (cid:107) p , since Q ∈ Q n − j and p ≥ p .Next we use estimate (2.8) (with M > d ) II ( x ) (cid:46) \ (cid:90) Q (cid:90) | y − w |≥ C M (2 j | y − w | ) − M | f j ( w ) | d w d y (cid:46) − jM (cid:107) f j (cid:107) p (cid:46) − nM (cid:107) f j (cid:107) p . We combine the estimates for I ( x ), II ( x ) and then, after taking suprema in x , in Q ∈ Q n − j ( x ) and in j , (5.8b) follows. (cid:3) The proof of Proposition 5.1 follows from (carefully) interpolating the twoprevious lemmas and a localization argument, as indicated in Figure 11.
Conclusion of the proof of Proposition 5.1.
We fix q > q . Observe that ρ min ( q ) − p = (1 − q q )(1 − p ) > p − ρ max ( q ) = (1 − q q ) p > ρ min ( q ) < p < ρ max ( q ). We first focus on the case ρ min ( q ) < p < ρ max ( q ),for which it suffices to consider r = p ; the corresponding inequality forsmaller r follows by H¨older’s inequality on [1 / , ρ max ( q ) ≤ p ≤ q will follow as a consequence of the previous range via alocalization argument. Let p be as in (5.4). In what follows set w ( j ) = 2 − jd and let (cid:96) qw ( L p ) bethe space of L p -valued sequences with (cid:107) F (cid:107) (cid:96) qw ( L p ) = (cid:16) (cid:88) j − jd (cid:107) f j (cid:107) qp (cid:17) /q . By linearization it suffices to consider, for any measurable choices of pos-itive integers x (cid:55)→ j ( x ) ∈ N , with j ( x ) ≥ n , cubes Q ( x ) ∈ Q n − j ( x ), andmeasurable L r (cid:48) ( R ) valued functions ( x, y ) (cid:55)→ v ( x, y, · ) in L ∞ ( R d ), the bi-linear operator M n [ F, v ]( x ) = \ (cid:90) Q ( x ) (cid:90) v ( x, y, s ) A j ( x ) f j ( x ) ( y, s ) d s d y and show that (cid:107)M n [ F, v ] (cid:107) L q (cid:46) − nε ( p,q,r ) (cid:107) F (cid:107) (cid:96) qw ( L p ) (cid:107) v (cid:107) L ∞ ( L r (cid:48) ) , (5.9)The conclusion for r ≤ p = p is immediate from Lemma 5.3, and inthe study of the range ρ min ( q ) < p < ρ max ( q ) we shall distinguish in whatfollows between the ρ min ( q ) < p < p and p < p < ρ max ( q ). The case ρ min ( q ) < p ≤ p . It suffices to prove (5.9) for r = p . We havefrom (5.6) (cid:107)M n [ F, v ] (cid:107) L q (cid:46) (cid:107) F (cid:107) (cid:96) qw ( L p ) (cid:107) v (cid:107) L ∞ ( L p (cid:48) ) for p = ρ min ( q ) (5.10)and from (5.7) (cid:107)M n [ F, v ] (cid:107) L q (cid:46) − nd ( q − q ) (cid:107) F (cid:107) (cid:96) qw ( L p ) (cid:107) v (cid:107) L ∞ ( L p (cid:48) ) . (5.11)One can interpolate (5.10) and (5.11), noting that for 0 < θ < ρ min ( q ) =1 − q q p (cid:48) , ρ min ( q ) < p < p − θp + θρ min ( q ) = p = ⇒ (1 − θ ) d ( q − q ) = dp (cid:48) q (cid:0) ρ min ( q ) − p (cid:1) , and deduce (cid:107)M n [ F, v ] (cid:107) L q (cid:46) − nε ( p,q,r ) (cid:107) F (cid:107) (cid:96) qw ( L p ) (cid:107) v (cid:107) L ∞ ( L p (cid:48) ) , for ρ min ( q ) ≤ p ≤ p , ε ( p, q, p ) = dp (cid:48) q (cid:0) ρ min ( q ) − p (cid:1) > j ( x ), Q ( x ). Thus wealso get (5.5) for r ≤ p , ρ min < p ≤ p and ε ( p, q, r ) = ε ( p, q, p ) as in (5.12).In order to carry out the interpolation argument one uses Stein’s inter-polation theorem on analytic families of operators, with an obvious analyticfamily suggested by the proof of the Riesz–Thorin theorem; we omit thedetails. Alternatively one can use Calder´on’s second method [ · , · ] θ , combin-ing a result on multilinear maps with a result on Banach lattices such as L ∞ ( X ), see [10, § § ARIATION BOUNDS FOR SPHERICAL AVERAGES 35
The case p < p < ρ max ( q ) . Again, it suffices to consider the case r = p .Note that for 0 < θ < ρ max ( q ) = q q p p < p < ρ max ( q ) − ϑp + ϑρ max ( q ) = p = ⇒ (1 − ϑ ) d ( q − q ) = dp q (cid:0) p − ρ max ( q ) (cid:1) (5.13)We claim (cid:107)M n [ F, v ] (cid:107) L q (cid:46) − nε ( p,q,p ) (cid:107) F (cid:107) (cid:96) qw ( L p ) (cid:107) v (cid:107) L ∞ ( L p (cid:48) ) , for p ≤ p ≤ ρ max ( q ) , ε ( p, q ) = dp q (cid:0) p − ρ max ( q ) (cid:1) > , (5.14)Given (5.13), the inequalities (5.14) can then be deduced by the above in-dicated interpolation arguments from (cid:107)M n [ F, v ] (cid:107) L q (cid:46) (cid:107) F (cid:107) (cid:96) qw ( L p ) (cid:107) v (cid:107) L ∞ ( L p (cid:48) ) for p = ρ max ( q ) (5.15)and (cid:107)M n [ F, v ] (cid:107) L q (cid:46) − nd ( q − q ) (cid:107) F (cid:107) (cid:96) qw ( L p ) (cid:107) v (cid:107) L ∞ ( L p (cid:48) ) . (5.16)Note that (5.15), (5.16) are immediate consequences of (5.6) and (5.7), re-spectively. The case ρ max ( q ) ≤ p ≤ q , < r < ρ max ( q ) . This case just follows bythe localization argument used in the proof of Lemma 4.1 via the kernelestimates (2.8), which allows to show that if (cid:107) M r,n F (cid:107) q (cid:46) − nε (cid:16) (cid:88) j ≥ n − jd (cid:107) f j (cid:107) qp ∗ (cid:17) /q holds for all 0 < r ≤ r ∗ and some 1 ≤ p ∗ ≤ q , then (cid:107) M r,n F (cid:107) q (cid:46) − nε (cid:16) (cid:88) j ≥ n − jd (cid:107) f j (cid:107) qp (cid:17) /q also holds for all p ∗ ≤ p ≤ q and all 0 < r ≤ r ∗ . For fixed q ∈ ( q , ∞ ],the desired estimates for ρ max ( q ) ≤ p ≤ q then follow from the above withinput inequalities r ∗ = p ∗ = ρ max ( q ) − (cid:15) for (cid:15) arbitrarily small. Note thatthis argument has already been used in the context of M r,n in the proof of(5.8b) in Lemma 5.3. We omit the details. (cid:3)
6. The sharp L p → L pd ( L p ) bound In this section we will give bounds for the sums of the operators A j which,in particular, will cover the crucial endpoint bound at P ( r ) = ( r , rd ) inTheorem 1.2, as well as the remaining endpoints bounds stated in Theorems1.2, 1.4 and 1.5. Proposition 6.1.
Let < p ≤ q < ∞ . Assume that sup j ≥ jd/q (cid:107)A j (cid:107) L p → L q ( L p ) ≤ C ≤ ∞ . (6.1) Let q < q < ∞ and define ρ max ( q ) = q q p and ρ min ( q ) = 1 − q q (1 − p ) .Assume that p , r are as in (5.4) , i.e. ρ min ( q ) < p ≤ q and (cid:40) r ≤ p if ρ min ( q ) < p < ρ max ( q ) ,r < ρ max ( q ) if ρ max ( q ) ≤ p ≤ q. Then for all { f j } j ≥ , (cid:13)(cid:13)(cid:13) (cid:88) j ≥ (cid:107)A j f j (cid:107) L r ( R ) (cid:13)(cid:13)(cid:13) L q ( R d ) ≤ C ( p, q )(1 + C ) (cid:16) (cid:88) j ≥ − jd (cid:107) f j (cid:107) qp (cid:17) /q . (6.2) Proof.
We first observe that by the monotone convergence theorem it sufficesto show (6.2) for any finite collection of functions { f j } n − j =0 , with uniformbounds in n ∈ N ; moreover, all f j can be assumed to be in the Schwartzclass. From Lemma 4.2 we get (cid:13)(cid:13)(cid:13) n − (cid:88) j =0 (cid:107)A j f j (cid:107) L r ( R ) (cid:13)(cid:13)(cid:13) L q ( R d ) (cid:46) n − /q (cid:16) (cid:88) j − jd (cid:107) f j (cid:107) qp (cid:17) /q , q ≤ q ≤ ∞ (6.3)and it is our task to remove the n -dependence in this estimate for p, q, r asin the statement of the Proposition.For a function G ∈ L q ( R d ) we recall the definition of the Fefferman–Steinsharp maximal function G ( x ) := sup x ∈ Q \ (cid:90) Q (cid:12)(cid:12)(cid:12) G ( y ) − \ (cid:90) Q G ( w ) d w (cid:12)(cid:12)(cid:12) d y which satisfies the bound (cid:107) G (cid:107) q ≤ c ( q ) (cid:107) G (cid:107) q for every q with q < q < ∞ ,and the implicit constant in this inequality is independent of the L q -normof G . This was proved in [15]. We may apply this inequality to G ( x ) = (cid:88) j ≥ (cid:16) (cid:90) |A j f j ( x, t ) | r d t (cid:17) /r , as its L q -norm is finite by (6.3); recall the sum is assumed to be finite. Let Q ( x ) = ∪ L ∈ Z Q L ( x ), i.e. the family of cubes containing x . We estimate G (cid:93) ( x ) (cid:46) G I ( x ) + G II ( x ) + G III ( x )where, with L ( Q ) as in (5.1), G I ( x ) := sup Q ∈Q ( x ) L ( Q ) ≤ \ (cid:90) Q (cid:12)(cid:12)(cid:12) (cid:88) ≤ j ≤− L ( Q ) (cid:16) (cid:107)A j f j ( y, · ) (cid:107) L r − \ (cid:90) Q (cid:107)A j f j ( w, · ) (cid:107) L r d w (cid:17)(cid:12)(cid:12)(cid:12) d y, G II ( x ) := sup Q ∈Q ( x ) L ( Q ) ≤ \ (cid:90) Q (cid:88) j ≥− L ( Q ) (cid:107)A j f j ( y, · ) (cid:107) L r d y, G III ( x ) := sup Q ∈Q ( x ) L ( Q ) > \ (cid:90) Q (cid:88) j ≥ (cid:107)A j f j ( y, · ) (cid:107) L r d y. ARIATION BOUNDS FOR SPHERICAL AVERAGES 37
The estimate for G III follows from the estimate for G II . To see this let U ( y ) = (cid:88) j ≥ (cid:107)A j f j ( y, · ) (cid:107) L r , U ∗ ( w ) = sup Q ∈Q ( w ) L ( Q )=0 \ (cid:90) Q U ( y ) d y. Given a cube (cid:101) Q ∈ Q ( x ) with L ( (cid:101) Q ) > (cid:101) Q into cubes of sidelength 1 and get \ (cid:90) (cid:101) Q U ( y ) d y ≤ \ (cid:90) (cid:101) Q U ∗ ( w ) d w ≤ M HL [ U ∗ ]( x )where M HL denotes the Hardy–Littlewood maximal operator. By a verycrude estimate we can replace U ∗ by G II and get G III ( x ) ≤ M HL [ G II ]( x ) . (6.4)The term G II is the most interesting but it has been already taken careof in §
5. We can write, with U j ( y ) := (cid:107)A j f j ( y, · ) (cid:107) L r G II ( x ) = sup Q ∈Q ( x ) L ( Q ) ≤ \ (cid:90) Q ∞ (cid:88) n =0 U − L ( Q )+ n ( y ) d y ≤ ∞ (cid:88) n =0 sup Q ∈Q ( x ) L ( Q ) ≤ \ (cid:90) Q U − L ( Q )+ n ( y ) d y = ∞ (cid:88) n =0 sup j ≥ n sup Q ∈Q n − j ( x ) \ (cid:90) Q U j ( y ) d y. Hence, with M r,n defined in (5.2) and F = { f j } j ≥ , we get G II ( x ) ≤ (cid:88) n ≥ M r,n F ( x ) . From Proposition 5.1, (6.4) and the L q -boundedness of the Hardy-Littlewoodmaximal operator we obtain (cid:107)G II (cid:107) q + (cid:107)G III (cid:107) q (cid:46) (cid:16) (cid:88) j ≥ − jd (cid:107) f j (cid:107) qp (cid:17) /q (6.5)for the range of ( p, q, r ) assumed in the proposition.It remains to consider the term G I , where we can use the cancellativeproperties of the (cid:107)G I (cid:107) q (cid:46) (cid:16) (cid:88) j ≥ − jd (cid:107) f j (cid:107) qp (cid:17) /q for ρ min ( q ) ≤ p ≤ q . (6.6)For n ≥ G I,n ( x ) := sup j ≥ sup Q ∈Q − n − j ( x ) \ (cid:90) Q (cid:12)(cid:12)(cid:12) (cid:107)A j f j ( y, · ) (cid:107) L r − \ (cid:90) Q (cid:107)A j f j ( w, · ) (cid:107) L r d w (cid:12)(cid:12)(cid:12) d y and, arguing as for G II , we observe that G I ( x ) ≤ (cid:80) n ≥ G I,n ( x ) . Our claim(6.6) will follow from the estimate (cid:107)G
I,n (cid:107) q (cid:46) − n (cid:16) (cid:88) j ≥ − jd (cid:107) f j (cid:107) qp (cid:17) /q for ρ min ( q ) ≤ p ≤ q (6.7)uniformly in n . In order to show this, note that by the triangle inequality G I,n ( x ) ≤ sup j ≥ sup Q ∈Q − n − j ( x ) \ (cid:90) Q \ (cid:90) Q (cid:13)(cid:13) A j f j ( y, · ) − A j f j ( w, · ) (cid:13)(cid:13) L r d w d y. Write A j f j = (cid:101) L j A j f j and let θ j be the convolution kernel of (cid:101) L j . Then for j ≤ n , x, y, w ∈ Q , Q ∈ Q − n − j (cid:16) (cid:90) (cid:12)(cid:12)(cid:12) A j f j ( y, t ) − A j f j ( w, t ) (cid:12)(cid:12)(cid:12) r d t (cid:17) /r ≤ (cid:16) (cid:90) (cid:12)(cid:12)(cid:12) (cid:90) ( θ j ( y − z ) − θ j ( w − z )) A j f j ( z, t ) d z (cid:12)(cid:12)(cid:12) r d t (cid:17) /r ≤ (cid:90) (cid:90) |(cid:104) y − w, ∇ θ j ( w + τ ( y − w ) − z ) (cid:105)| d τ (cid:16) (cid:90) (cid:12)(cid:12) A j f j ( z, t ) (cid:12)(cid:12) r d t (cid:17) /r d z. Since 1 + 2 j | w + τ ( y − w ) − z | ≈ j | x − z | for x, y, w ∈ Q , Q ∈ Q − n − j ( x ), τ ∈ [0 , C N − n − j (cid:90) j ( d +1) (1 + 2 j | x − z | ) N (cid:16) (cid:90) |A j f j ( z, t ) | r d t (cid:17) /r d z and hence we get for n ≥ N > d |G I,n ( x ) | (cid:46) − n M HL (cid:2) sup j ≥ (cid:107)A j f j (cid:107) L r ( R ) (cid:3) ( x ) . This implies, using the L q boundedness of the Hardy–Littlewood maximaloperator M HL and Lemma 4.2, (cid:107)G I,n (cid:107) L q (cid:46) − n (cid:13)(cid:13) sup j ≥ (cid:107)A j f j (cid:107) L r ( R ) (cid:13)(cid:13) L q (cid:46) − n (cid:13)(cid:13)(cid:13)(cid:16) (cid:88) j ≥ (cid:107)A j f j (cid:107) qL r ( R ) (cid:17) /q (cid:13)(cid:13)(cid:13) L q = 2 − n (cid:16) (cid:88) j ≥ (cid:107)A j f j (cid:107) qL q ( L r ) (cid:17) /q (cid:46) − n (cid:16) (cid:88) j ≥ − jd (cid:107) f j (cid:107) qp (cid:17) /q , which is (6.7) and thus (6.6) is proved. The proof is complete after combining(6.6) and (6.5). (cid:3) As (6.1) holds with p = 2, q = d +1) d − by Lemma 2.4, Proposition 6.1yields the following. Corollary 6.2.
Assume that d +1) d − < q < ∞ , d +1 d − q < p < − d +1 d − q , r ≤ p (6.8) or d +1) d − < q < ∞ , q ≤ p ≤ d +1 d − q , r < q ( d − d +1 . (6.9) ARIATION BOUNDS FOR SPHERICAL AVERAGES 39
Then for all { f j } j ≥ , (cid:13)(cid:13)(cid:13) (cid:88) j ≥ (cid:107)A j f j (cid:107) L r ( R ) (cid:13)(cid:13)(cid:13) L q ( R d ) (cid:46) (cid:16) (cid:88) j ≥ − jd (cid:107) f j (cid:107) qp (cid:17) /q . Combining this with Lemma 2.7 one obtains the strong type bound at P ( r ) in Theorem 1.2 (that is, Corollary 1.3). Proposition 6.3.
Let d ≥ , d +1 d ( d − < p < ∞ . Then (cid:107)A f (cid:107) L pd ( B /pp, ) ≤ (cid:107) f (cid:107) L p . (6.10) Moreover, (cid:107) V Ip Af (cid:107) L pd ≤ (cid:107) f (cid:107) L p . (6.11) Proof.
The bound (6.11) follows from (6.10) via (2.4).To prove (6.10) we apply Corollary 6.2 and Lemma 2.7 with s = d/q =1 /p . We verify the assumption (6.8) for q = pd , r = p . The condition d +1 d − q < p is satisfied (for q = pd ) when d − d − >
0, which holds when d ≥
3. The condition p < − d +1 d − q is satisfied (for q = pd ) if p > d +1 d ( d − .The condition q > d +1) d − is also satisfied if q = pd , p > d +1 d ( d − . In particular,the latter implies that q = pd > (cid:3) Arguing in a similar way, we obtain the remaining claimed endpointbounds in Theorems 1.2, 1.4 and 1.5.
Proposition 6.4.
Let d ≥ and r > d +1 d ( d − .(i) Let r ≤ p ≤ rd . Then the operators A : L p → L rd ( B /rr, ) and V Ir A : L p → L rd are bounded. That is, V Ir : L p → L q is bounded if (1 /p, /q ) belongs to the closed segment [ Q ( r ) , P ( r )] in Theorem 1.2.(ii) Let d +1 d ( d − < p ≤ r . Then the operators A : L p → L pd ( B /rr, ) and V Ir A : L p → L pd are bounded. That is, V Ir : L p → L q is bounded if (1 /p, /q ) belongs to the half-open segment [ P ( r ) , Q ) in Theorem 1.2.Proof. For part (i) we use again Corollary 6.2 and Lemma 2.7 with s = d/q =1 /r . The condition (6.8) yields the ranges rd ( d − rd ( d − − ( d +1) < p < rd ( d − d +1 and r ≤ p . Note that r > rd ( d − rd ( d − − ( d +1) if and only if r > d +1 d ( d − ; moreover,the condition q > d +1) d − (for q = rd ) is satisfied in the range r > d +1 d ( d − whenever d − d − >
0, which holds for d ≥
3. This settles the range r ≤ p < rd ( d − d +1 . The range rd ( d − d +1 ≤ p ≤ rd corresponds to (6.9). Notethat the condition r < q ( d − d +1 requires (for q = rd ) d − d − >
0, whichholds when d ≥
3. The condition q > d +1) d − was already verified for (6.8).This concludes the bounds in (i). Part (ii) follows from standard embedding theorems from Proposition6.3. (cid:3)
Proposition 6.5.
Let d ≥ and ≤ r ≤ d +1 d ( d − or d = 3 and / < r ≤ / . Let rd ( d − rd ( d − − d − < p ≤ rd . Then the operators A : L p → L rd ( B /rr, ) and V Ir A : L p → L rd are bounded. That is, V Ir : L p → L q is bounded if (1 /p, /q ) belongs to the half-open segment [ Q ( r ) , Q ( r )) in Theorems 1.4 and 1.5.Proof. This follows arguing as in the proof of Proposition 6.4. The onlydifference is that in (6.8) the relevant range for p (for q = rd ) if r ≤ d +1 d ( d − is rd ( d − rd ( d − − d − < p < rd ( d − d +1 . Moreover, the condition q > d +1) d − requires (for q = rd ) that r > d +1) d ( d − . As r ≥
1, this condition is satisfied if d − d − > d ≥
4. If d = 3, we require the restriction r > / (cid:3)
7. An endpoint bound for the global variation
The purpose of this section is to prove Theorem 1.1.It will be useful to work with the standard homogeneous Littlewood–Paleydecomposition. We define P j f , j ∈ Z by (cid:100) P j f ( ξ ) = (cid:0) β (2 − j | ξ | ) − β (2 − j | ξ | ) (cid:1) (cid:98) f ( ξ )so that P j localizes to frequencies of size ≈ j . We have P j = L j for j ≥ L f = (cid:80) j ≤ P j f for f ∈ L p , p ∈ (1 , ∞ ) with convergence in L p . Itwill also be convenient to use reproducing operators (cid:101) P j localizing to slightlylarger frequency annuli, with (cid:101) P j P j = P j for j ∈ Z .Let j ≥
0. We recall the definition A j f ( x, t ) = χ ( t ) A t L j f ( x ) = χ ( t ) K j,t ∗ f ( x ) where (cid:100) K j,t ( ξ ) = (cid:98) σ ( tξ ) β j ( | ξ | ) . We combine this with dyadic dilations, and define for k ∈ Z , t ∈ [1 / , A kj f ( x, t ) = χ ( t ) A k t P j − k f ( x ) = χ ( t ) K kj,t ∗ f ( x ) (7.1)where K kj,t = 2 − kd K j,t (2 − k · ). Below we shall often use a scaled version of(2.8), namely | K kj,t ( x ) | (cid:46) N − kd (2 j − k | x | ) − N , | x | ≥ · k , t ∈ [1 / , . (7.2)We start recording the following special case of Proposition 4.3, whichwill be relevant for the proof of Theorem 1.1 (when p = q ). Corollary 7.1.
For ≤ r ≤ ∞ , r ( d +1) d − ≤ q ≤ ∞ , r ≤ p ≤ q we have (cid:107)A j f (cid:107) L q ( L r ) (cid:46) − jd/q (cid:107) f (cid:107) p . By Corollary 7.1 and rescaling we have (cid:107)A kj (cid:107) L p → L p ( L r ) (cid:46) − jd/p , r ( d +1) d − ≤ p ≤ ∞ . (7.3)One can improve over this result and extend it to sums in k whenever r ( d +1) d − < p < ∞ . ARIATION BOUNDS FOR SPHERICAL AVERAGES 41
Proposition 7.2.
For ≤ r < ∞ , r ( d +1) d − < p < ∞ we have, for all j ≥ , (cid:13)(cid:13)(cid:13)(cid:16) (cid:88) k ∈ Z (cid:107)A kj f (cid:107) rL r ( R ) (cid:17) /r (cid:13)(cid:13)(cid:13) L p ( R d ) (cid:46) − jd/p (cid:107) f (cid:107) p . (7.4) Proof.
As in the proof of Proposition 6.1, by the monotone convergencetheorem, it suffices to show (7.4) for any finite collection {A kj } k ∈ K , withuniform bounds on the cardinality of the finite set K ⊂ Z .We use again the sharp maximal function of Fefferman–Stein. Let G ( x ) = (cid:0) (cid:88) k ∈ K (cid:107)A kj f ( x, · ) (cid:107) rL r (cid:1) /r , which has finite L p norm whenever p = r ( d +1) d − ; note that (7.3) and Minkowski’sinequality imply that (cid:107) G (cid:107) p (cid:46) | K | /r − jd/p (cid:107) f (cid:107) p . By the bound (cid:107) G (cid:107) p (cid:46) p (cid:107) G (cid:107) p , it suffices to show (cid:107) G (cid:107) p (cid:46) − jd/p (cid:107) f (cid:107) p ,uniformly on the finite set K for p < p < ∞ . By the triangle inequality,we dominate G ( x ) ≤ sup Q ∈Q ( x ) \ (cid:90) Q \ (cid:90) Q (cid:16) (cid:88) k ∈ Z (cid:107)A kj f ( y, · ) − A kj f ( w, · ) (cid:107) rL r (cid:17) /r d w d y ≤ G f ( x ) + ∞ (cid:88) n =1 U n f ( x )where G f ( x ) := sup L ∈ Z sup Q ∈Q L ( x ) \ (cid:90) Q (cid:16) (cid:88) k ≤ L (cid:107)A kj f ( y, · ) (cid:107) rL r (cid:17) /r d y and U n f ( x ) := sup L ∈ Z sup Q ∈Q L ( x ) \ (cid:90) Q \ (cid:90) Q (cid:13)(cid:13) A L + nj f ( y, · ) − A L + nj f ( w, · ) (cid:13)(cid:13) L r d w d y. We claim that for r ( d +1) d − ≤ p ≤ ∞ , 2 ≤ r < ∞ we have (cid:107)G f (cid:107) p (cid:46) − jα ( r ) (cid:107) f (cid:107) p , for some α ( r ) > d/p (7.5)and (cid:107)U n f (cid:107) p (cid:46) (cid:40) ( n − j ) d ( d − r ( d +1) − p ) − jd/p (cid:107) f (cid:107) p , if 1 ≤ n ≤ j ,2 j − n − jd/p (cid:107) f (cid:107) p , if n > j . (7.6)The desired bound (cid:107) G (cid:107) p (cid:46) − jd/p (cid:107) f (cid:107) p follows summing in n if p > r ( d +1) d − . Proof of (7.5) . We prove inequalities for G on L r and L ∞ which will yield(7.5) by interpolation.Let p = r ( d +1) d − . We first observe the inequalities (cid:107)A kj f (cid:107) L ( L r ) (cid:46) − j ( d − + r ) (cid:107) f (cid:107) L (7.7) (cid:107)A kj f (cid:107) L p ( L r ) (cid:46) − jd/p (cid:107) f (cid:107) L p (7.8)uniformly in k . The estimate (7.7) holds from the bounds (cid:107)A kj (cid:107) L → L ( L ) (cid:46) − j ( d − / and (cid:107) ∂ t A kj (cid:107) L → L ( L ) (cid:46) − j ( d − / (which follow from (2.13)) andthe Sobolev embedding theorem, and (7.8) is (7.3) with p = p . Since2 ≤ r < p and d − + r > dp one has by interpolation that (cid:107)A kj f (cid:107) L r ( L r ) (cid:46) − jα ( r ) (cid:107) f (cid:107) L r , for some α ( r ) > d/p . (7.9)This implies (cid:107)G f (cid:107) L r (cid:46) (cid:13)(cid:13)(cid:13) M HL (cid:2) ( (cid:88) k ∈ Z (cid:107)A kj (cid:101) P j − k f (cid:107) rL r ( R ) ) /r (cid:3)(cid:13)(cid:13)(cid:13) r (cid:46) (cid:16) (cid:88) k ∈ Z (cid:107)A kj (cid:101) P j − k f (cid:107) rL r ( L r ) (cid:17) /r (cid:46) − jα ( r ) (cid:16) (cid:88) k ∈ Z (cid:107) (cid:101) P j − k f (cid:107) rr (cid:17) /r (cid:46) − jα ( r ) (cid:107) f (cid:107) r (7.10)by Littlewood–Paley theory, since r ≥ L ∞ bound. Fix x , L , Q ∈ Q L ( x ) and let B Lx be theball centered at x with radius d L +10 . Using H¨older’s inequality and theembedding (cid:96) ⊆ (cid:96) r for r ≥ \ (cid:90) Q (cid:16) (cid:88) k ≤ L (cid:13)(cid:13) A kj f ( y, · ) (cid:13)(cid:13) rL r (cid:17) /r d y ≤ I ( x ) + II ( x )where I ( x ) = (cid:16) \ (cid:90) Q (cid:88) k ≤ L (cid:13)(cid:13) A kj [ B Lx f ]( y, · ) (cid:13)(cid:13) rL r d y (cid:17) /r II ( x ) = \ (cid:90) Q (cid:88) k ≤ L (cid:13)(cid:13) A kj [ R d \ B Lx f ]( y, · ) (cid:13)(cid:13) L r d y. We have, in view of A kj = A kj (cid:101) P j − k and using (7.9), I ( x ) (cid:46) − Ld/r (cid:16) (cid:88) k ≤ L (cid:13)(cid:13) A kj (cid:101) P j − k [ B Lx f ] (cid:13)(cid:13) rL r ( L r ) (cid:17) /r (cid:46) − jα ( r ) − Ld/r (cid:16) (cid:88) k ≤ L (cid:13)(cid:13) (cid:101) P j − k [ B Lx f ] (cid:13)(cid:13) rL r (cid:17) /r (cid:46) − jα ( r ) − Ld/r (cid:107) B Lx f (cid:107) L r (cid:46) − jα ( r ) (cid:107) f (cid:107) L ∞ , ARIATION BOUNDS FOR SPHERICAL AVERAGES 43 using r ≥ II ( x ) we use (7.2) and estimate II ( x ) (cid:46) N (cid:88) k ≤ L \ (cid:90) Q (cid:90) R d \ B Lx − kd (2 j − k | y − w | ) N | f ( w ) | d w d y (cid:46) − jN (cid:88) k ≤ L ( L − k )( d − N ) (cid:107) f (cid:107) ∞ (cid:46) − jN (cid:107) f (cid:107) ∞ , where N > d . We combine the estimates for I ( x ) and II ( x ) to obtain (cid:107)G f (cid:107) ∞ (cid:46) − jα ( r ) (cid:107) f (cid:107) ∞ . (7.11)Interpolating (7.10) and (7.11) and noting that that α ( r ) > d/p ≥ d/p for p ≥ p we obtain (7.5). Proof of (7.6) for ≤ n ≤ j . This case is similar to that of G . Let p = r ( d +1) d − . We get the asserted estimate by interpolating between theinequalities (cid:107)U n f (cid:107) p (cid:46) − jd/p (cid:107) f (cid:107) p (7.12) (cid:107)U n f (cid:107) ∞ (cid:46) ( n − j ) d/p (cid:107) f (cid:107) ∞ . (7.13)To see (7.12), we use A kj = A kj (cid:101) P j − k and (7.8) to estimate (cid:107)U n f (cid:107) p (cid:46) (cid:13)(cid:13)(cid:13) M HL (cid:2)(cid:0) (cid:88) k ∈ Z (cid:107)A kj (cid:101) P j − k f (cid:107) p L r ( R ) ) /p (cid:3)(cid:13)(cid:13)(cid:13) p (cid:46) (cid:16) (cid:88) k ∈ Z (cid:107)A kj (cid:101) P j − k f (cid:107) p L p ( L r ) ) /p (cid:46) − jd/p (cid:0) (cid:88) k ∈ Z (cid:107) (cid:101) P j − k f (cid:107) p p ) /p (cid:46) − jd/p (cid:107) f (cid:107) p , using that p ≥ x , L , Q ∈ Q L ( x ), let B L + nx be the ball centered at x with radius d L + n +10 and estimate \ (cid:90) Q \ (cid:90) Q (cid:13)(cid:13) A L + nj f ( y, · ) − A L + nj f ( w, · ) (cid:13)(cid:13) L r d w d y (cid:46) \ (cid:90) Q (cid:13)(cid:13) A L + nj (cid:101) P j − L − n f ( y, · ) (cid:13)(cid:13) L r d y ≤ III ( x ) + IV ( x )where III ( x ) = (cid:16) \ (cid:90) Q (cid:13)(cid:13) A L + nj [ B L + nx f ]( y, · ) (cid:13)(cid:13) p L r d y (cid:17) /p ,IV ( x ) = \ (cid:90) Q (cid:13)(cid:13) A L + nj [ R d \ B L + nx f ]( y, · ) (cid:13)(cid:13) L r d y. We get by (7.3)
III ( x ) (cid:46) − Ld/p − jd/p (cid:107) B L + nx f (cid:107) p (cid:46) ( n − j ) d/p (cid:107) f (cid:107) ∞ . Moreover, by (7.2), IV ( x ) (cid:46) \ (cid:90) Q (cid:90) R d \ B L + nx − ( L + n ) d (2 j − L − n | y − w | ) N | f ( w ) | d w d y (cid:46) − jN (cid:107) f (cid:107) ∞ for any N > d . The estimates for
III ( x ) and IV ( x ) yield (7.13) for 1 ≤ n ≤ j . Proof of (7.6) for n > j . Here we use cancellation and note that for x ∈ Q \ (cid:90) Q \ (cid:90) Q (cid:13)(cid:13) (cid:101) P j − L ( Q ) − n g ( y, · ) − (cid:101) P j − L ( Q ) − n g ( w, · ) (cid:13)(cid:13) L r d w d y (cid:46) j − n M HL [ (cid:107) g (cid:107) L r ( R ) ]( x ) . Using this with g = A L + nj f = (cid:101) P j − L − n A L + nj and the Fefferman–Stein in-equality for sequences of Hardy–Littlewood maximal functions, we may thenestimate (cid:13)(cid:13)(cid:13) sup L ∈ Z sup Q ∈Q L ( x ) \ (cid:90) Q \ (cid:90) Q (cid:13)(cid:13) A L + nj f ( y, · ) − A L + nj f ( w, · ) (cid:13)(cid:13) L r d w d y (cid:13)(cid:13)(cid:13) L p ( dx ) (cid:46) j − n (cid:13)(cid:13)(cid:13) sup k ∈ Z M HL (cid:2) (cid:107)A kj f (cid:107) L r ( R ) (cid:3)(cid:13)(cid:13)(cid:13) p (cid:46) j − n (cid:16) (cid:88) k ∈ Z (cid:13)(cid:13) A kj (cid:101) P j − k f (cid:107) pL p ( L r ) (cid:17) /p (cid:46) j − n − jd/p (cid:16) (cid:88) k ∈ Z (cid:107) (cid:101) P j − k f (cid:107) pp (cid:17) /p (cid:46) j − n − jd/p (cid:107) f (cid:107) p for r ( d +1) d − ≤ p ≤ ∞ , using (7.3) in the third inequality and p ≥ n > j .This finishes the proof of the proposition. (cid:3) Remark.
The difficulty for putting the pieces together comes because it isassumed r ( d +1) d − < p . If one had r ≥ p , one can simply put pieces togetherby standard Littlewood–Paley theory as, for instance, in (7.10).A consequence of Proposition 7.2 is the following restricted weak typebound. Proposition 7.3.
For d ≥ , r ≥ , (cid:13)(cid:13)(cid:13)(cid:16) (cid:88) k ∈ Z (cid:13)(cid:13)(cid:13) (cid:88) j ≥ A kj f (cid:13)(cid:13)(cid:13) rB /rr, ( R ) (cid:17) /r (cid:13)(cid:13)(cid:13) L rd, ∞ ( R d ) (cid:46) (cid:107) f (cid:107) L rd, ( R d ) . (7.14) Proof.
WriteΛ l A kj f ( x, t ) = 2 − j ( d − / (2 π ) − ( d +1) (cid:88) ± (cid:90) − kd κ ± j,l (2 − k y, t ) f ( x − y ) dy where κ ± j,l is as in (2.17). ARIATION BOUNDS FOR SPHERICAL AVERAGES 45
We first show that for all j ≥
1, 2 ≤ r < ∞ , r ( d +1) d − < p < ∞ , (cid:13)(cid:13)(cid:13)(cid:16) (cid:88) k ∈ Z (cid:107)A kj f (cid:107) rB /rr, ( R ) (cid:17) /r (cid:13)(cid:13)(cid:13) L p ( R d ) (cid:46) − j ( d/p − /r ) (cid:107) f (cid:107) p . (7.15)Indeed, by Proposition 7.2, for | j − l | ≤ (cid:13)(cid:13)(cid:13)(cid:16) (cid:88) k ∈ Z l/r (cid:107) Λ l A kj f (cid:107) rL r ( R ) (cid:17) /r (cid:13)(cid:13)(cid:13) L p ( R d ) (cid:46) − j ( d/p − /r ) (cid:107) f (cid:107) p . (7.16)Moreover for | j − l | ≥
10, we get from (2.18) (cid:13)(cid:13)(cid:13)(cid:16) (cid:88) k ∈ Z l/r (cid:107) Λ l A kj f (cid:107) rL r ( R ) (cid:17) /r (cid:13)(cid:13)(cid:13) L p ( R d ) (cid:46) N min { − j ( N − r ) , − l ( N − r ) } (cid:13)(cid:13)(cid:13)(cid:16) (cid:88) k ∈ Z (cid:12)(cid:12) M HL [ (cid:101) P j − k f ] | r (cid:17) /r (cid:13)(cid:13)(cid:13) p (cid:46) N min { − j ( N − r ) , − l ( N − r ) }(cid:107) f (cid:107) p , (7.17)using that the Fefferman–Stein and Littlewood–Paley inequalities togetherimply (cid:13)(cid:13)(cid:13)(cid:16) (cid:88) k ∈ Z (cid:12)(cid:12) M HL [ (cid:101) P j − k f ] | r (cid:17) /r (cid:13)(cid:13)(cid:13) p (cid:46) p (cid:107) f (cid:107) p , < p < ∞ , r ≥ . Then (7.15) follows summing over (cid:96) ≥ § A j f ( x ) := (cid:16) (cid:88) k ∈ Z (cid:107)A kj f ( x, · ) (cid:107) rB /rr, ( R ) (cid:17) /r . Note that r ( d +1) d − < rd when d − d − >
0, that is, d ≥
3. Let p , p be such that r ( d +1) d − < p < rd < p . By (7.15) we have that (cid:107) A j f (cid:107) p i (cid:46) − j ( d/p i − /r ) (cid:107) f (cid:107) p i , i = 0 , rd, rd ) inequalityfor (cid:80) j ≥ A j follows from Lemma 2.8. This implies the assertion. (cid:3) Conclusion of the proof of Theorem 1.1.
Following [21], we write V r Af ( x ) ≤ V dyad r Af ( x ) + V sh r Af ( x )where V dyad r Af ( x ) := sup N ∈ N sup k < ···
2] we get V I k r Af ( x ) ≤ (cid:12)(cid:12)(cid:12) ∞ (cid:88) j =0 A kj f ( x, · ) (cid:12)(cid:12)(cid:12) V r ( I ) by the definition of A kj in (7.1). The term corresponding to j = 0 is easilyestimated by a square function (cid:16) (cid:88) k ∈ Z |A k f ( x, · ) | rV r ( I ) (cid:17) /r (cid:46) (cid:16) (cid:88) k ∈ Z (cid:90) (cid:12)(cid:12) ddt A k f ( x, t ) (cid:12)(cid:12) d t (cid:17) / . We claim for 1 < p < ∞ (cid:13)(cid:13)(cid:13)(cid:16) (cid:88) k ∈ Z (cid:90) (cid:12)(cid:12) ddt A k f ( x, t ) (cid:12)(cid:12) d t (cid:17) / (cid:13)(cid:13)(cid:13) p ≤ C p (cid:107) f (cid:107) p . (7.18)Since χ (cid:48) ( t ) = 0 for 1 ≤ t ≤ ddt (cid:100) A k f ( ξ, t ) = χ ( t ) (cid:104) k ξ, ∇ (cid:98) σ (2 k tξ ) (cid:105) β (2 k | ξ | ) (cid:98) f ( ξ )Using Plancherel’s theorem and interchanging sums and integrals one gets(7.18) for p = 2. We then invoke standard Calder´on–Zygmund theory in theHilbert-space setting (see [42, ch. II.5]) to see that (7.18) holds in the fullrange 1 < p < ∞ . It follows that for r ≥ (cid:13)(cid:13)(cid:13)(cid:16) (cid:88) k ∈ Z |A k f ( x, · ) | rV r ( I ) (cid:17) /r (cid:13)(cid:13)(cid:13) p (cid:46) p (cid:107) f (cid:107) p which is stronger than the required L p, → L p, ∞ bound.It remains to consider the cases j ≥
1. By the embedding (2.4) we have (cid:16) (cid:88) k ∈ Z (cid:12)(cid:12)(cid:12) (cid:88) j ≥ A kj f ( x, · ) (cid:12)(cid:12)(cid:12) rV r (cid:17) /r (cid:46) (cid:16) (cid:88) k ∈ Z (cid:13)(cid:13)(cid:13) (cid:88) j ≥ A kj f ( x, · ) (cid:13)(cid:13)(cid:13) rB /rr, (cid:17) /r . We apply the restricted weak type inequality of Proposition 7.3 to the ex-pression on the right-hand side to conclude the desired bound for V sh r . Thisfinishes the proof. (cid:3) Remark.
If in two dimensions one has the conjectured local smoothing end-point results for p > r, r ) estimate (7.14) for r >
2. The conjectured endpoint estimate in theassumptions seems currently out of reach.
ARIATION BOUNDS FOR SPHERICAL AVERAGES 47
8. A sparse domination result
We conclude the paper with a discussion of the sparse domination resultfor the global V r A in Theorem 1.7. It is indeed an immediate consequence ofa special case of a result on convolution operators with compactly supporteddistributions which can be found in [2, Prop.7.2].We let u be a compactly supported distribution, define the dilate in thesense of distributions by (cid:104) u t , f (cid:105) = (cid:104) u, f ( t · ) (cid:105) and let T f ( x, t ) = f ∗ u t . Forfixed x let V r T f denote the r -variation norm of t (cid:55)→ T f ( x, t ). As before let I = [1 ,
2] and V Ir f ( x ) the corresponding variation norm over I . Theorem 8.1. [2] . Let < p ≤ q < ∞ , and u ∈ S (cid:48) ( R d ) with compactsupport in R d \ { } . (i) Suppose that (cid:107) V r T (cid:107) L p → L p, ∞ + (cid:107) V r T (cid:107) L q, → L q < ∞ , (8.1) (cid:107) V Ir T (cid:107) L p → L q < ∞ , (8.2) and that there is an ε > so that for all λ ≥ , and all Schwartz function f with supp (cid:98) f ⊂ { ξ : λ/ < | ξ | < λ } , (cid:107) V Ir T f (cid:107) q ≤ Cλ − ε (cid:107) f (cid:107) p . (8.3) Then there is a constant C = C ( p, q ) such that for each pair of compactlysupported bounded functions f , f there is a sparse family of cubes S ( f , f ) such that (cid:90) V r T f ( x ) f ( x ) dx ≤ C (cid:88) Q ∈ S ( f ,f ) | Q |(cid:104) f (cid:105) Q,p (cid:104) f (cid:105) Q,q (cid:48) . (8.4) (ii) Suppose that in addition p < q , and suppose that (8.4) holds with aconstant independently of f , f . Then conditions (8.1) , (8.2) hold.Proof of Theorem 1.7. We let u be surface measure on the unit sphere. Asdiscussed in the introduction the inequalities in (8.1) were already provedin the relevant ranges of Theorem 1.7 in [21]. The inequalities (8.2) and(8.3) in the asserted ranges follow from the single-scale frequency boundsin Propositions 4.6 and 4.7. Thus the sparse bounds in Theorem 1.7 are aconsequence of part (i) of Theorem 8.1. The sharpness of the sparse boundsfollows from part (ii); see also § (cid:3) References [1] AimPL,
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David Beltran: Department of Mathematics, University of Wisconsin, 480Lincoln Drive, Madison, WI, 53706, USA.
E-mail address : [email protected] Richard Oberlin: Department of Mathematics, Florida State University,Tallahassee, FL 32306-4510, USA
E-mail address : [email protected] Luz Roncal: Basque Center for Applied Mathematics (BCAM), 48009, Bil-bao, Spain and Ikerbasque, Basque Foundation for Science, 48011 Bilbao, Spain
E-mail address : [email protected] Andreas Seeger: Department of Mathematics, University of Wisconsin, 480Lincoln Drive, Madison, WI, 53706, USA.
E-mail address : [email protected] Betsy Stovall: Department of Mathematics, University of Wisconsin, 480Lincoln Drive, Madison, WI, 53706, USA.
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