On the invariance of certain vanishing subspaces of Morrey spaces with respect to some classical operators
aa r X i v : . [ m a t h . F A ] N ov ON THE INVARIANCE OF CERTAIN VANISHING SUBSPACES OFMORREY SPACES WITH RESPECT TO SOME CLASSICALOPERATORS
AYSEGUL Ç. ALABALIK, ALEXANDRE ALMEIDA ∗ , AND STEFAN SAMKO Abstract.
We consider subspaces of Morrey spaces defined in terms of various vanish-ing properties of functions. Such subspaces were recently used to describe the closure of C ∞ ( R n ) in Morrey norm. We show that these subspaces are invariant with respect tosome classical operators of harmonic analysis, such as the Hardy-Littlewood maximaloperator, singular type operators and Hardy operators. We also show that the vanish-ing properties defining those subspaces are preserved under the action of Riesz potentialoperators and fractional maximal operators. Introduction
Morrey spaces play an important role in the study of local behaviour and regularityproperties of solutions to PDE, including heat equations and Navier-Stokes equations.We refer to [16, 18, 37, 40, 41] and references therein for further details. For ≤ p < ∞ , ≤ λ ≤ n , the classical Morrey space L p,λ ( R n ) consists of all locally p -integrable functions f on R n with finite norm(1.1) k f k p,λ := sup x ∈ R n , r> M p,λ ( f ; x, r ) /p , where(1.2) M p,λ ( f ; x, r ) := 1 r λ Z B ( x,r ) | f ( y ) | p dy , x ∈ R n , r > . Straightforward calculations show that k f ( t · ) k L p,λ ( R n ) = t λ − np k f k L p,λ ( R n ) , t > , which implies a modification of the scaling factor in comparison with L p -spaces.It is well known that the spaces L p,λ ( R n ) are non-separable if λ > (see [31, Propo-sition 3.7] for a proof). The lack of approximation tools for the entire Morrey spacehas motivated the introduction of appropriate subspaces, like the Zorko space ([45]) col-lecting all Morrey functions for which the translation is continuous in Morrey norm andvanishing spaces defined in terms of the vanishing properties ( V ), ( V ∞ ) and ( V ∗ ) definedbelow. Date : March 20, 2019.2010
Mathematics Subject Classification.
Key words and phrases.
Morrey spaces, vanishing properties, maximal functions, potential operators,singular operators, Hardy operators. ∗ Corresponding author.A. Almeida was partially supported by the Portuguese national funding agency for science, researchand technology (FCT), within the Center for Research and Development in Mathematics and Applica-tions (CIDMA), project UID/MAT/04106/2019. S. Samko was supported by Russian Foundation forBasic Research under the grants 19-01-00223 and 18-01-00094-a.
The theory of Morrey spaces goes back to Morrey [21] who considered related integralinequalities in the study of solutions to nonlinear elliptic equations. In the form of Banachspaces of functions, called thereafter Morrey spaces, the ideas of Morrey [21] were furtherdeveloped by Campanato [10] and Peetre [23]. We refer to the books [2, 14, 27, 39, 40]and the overview [28] for additional references and basic properties of these spaces andsome of their generalizations. A discussion on Harmonic Analysis in Morrey spaces canbe found in [3], [30], [41].Many classical operators from Harmonic Analysis such as maximal operators, singularoperators, potential operators and Hardy operators, are known to be bounded in Morreyspaces. There are many papers in the literature dealing with this subject, including thecase when the spaces and/or the operators have generalized parameters. We refer to thepapers [1, 4, 7, 8, 11, 15, 16, 17, 19, 20, 22, 24, 30, 34, 35, 36, 38].In this paper we are interested in studying the behavior of those classical operatorsin certain subspaces of Morrey spaces. We consider the following subspaces of L p,λ ( R n ) .The class V L p,λ ( R n ) consists of all those functions f ∈ L p,λ ( R n ) such that( V ) lim r → sup x ∈ R n M p,λ ( f ; x, r ) = 0 . Similarly, V ∞ L p,λ ( R n ) is the set of all f ∈ L p,λ ( R n ) such that( V ∞ ) lim r →∞ sup x ∈ R n M p,λ ( f ; x, r ) = 0 . We also consider the set V ( ∗ ) L p,λ ( R n ) consisting of all functions f ∈ L p,λ ( R n ) having thevanishing property( V ∗ ) lim N →∞ A N,p ( f ) := lim N →∞ sup x ∈ R n Z B ( x, | f ( y ) | p χ N ( y ) dy = 0 , where χ N := χ R n \ B (0 ,N ) , N ∈ N . The three vanishing classes defined above are closed sets in L p,λ ( R n ) with respect to thenorm (1.1). The space V L p,λ ( R n ) , often called in the literature just by vanishing Morreyspace , was already introduced in [12, 42, 43] in connection with applications to PDE. Thesubspaces V ∞ L p,λ ( R n ) and V ( ∗ ) L p,λ ( R n ) were recently introduced in [5, 6] to study theapproximation problem by nice functions in Morrey spaces. Note that V ∞ L p,λ ( R n ) wasindependently considered in [44] in the study of interpolation problems.The subspace V ( ∗ )0 , ∞ L p,λ ( R n ) , collecting those Morrey functions having all the vanishingproperties ( V ), ( V ∞ ) and ( V ∗ ), provides an explicit description of the closure of C ∞ ( R n ) in Morrey norm, see [5, Theorem 5.3 and Corollary 5.4].The boundedness of classical operators in vanishing Morrey spaces at the origin wasalready studied in some papers, including the case of generalized parameters, see [25, 29,32, 33]. In particular, in [32] it was studied a class of sublinear singular type operatorswhich includes the Hardy-Littlewood maximal function and Calderón-Zygmund operatorswith standard kernels. The boundedness results in [32] were given in terms of Zygmund-type integral conditions on the function parameter defining the Morrey space.Up to authors’ knowledge, the boundedness of classical operators in the vanishingMorrey spaces V ∞ L p,λ ( R n ) and V ( ∗ ) L p,λ ( R n ) was not touched so far, apart some resultsin [5, Theorem 3.8], [6, Corollary 4.3] where it was observed that convolution operatorswith integrable kernels are bounded in those subspaces. It is the main goal of this paperto show that the vanishing properties defining these subspaces are preserved under theaction of many other operators from Harmonic Analysis, including maximal, singular, NVARIANCE OF VANISHING MORREY SUBSPACES 3 potential and Hardy operators. One of the key results is the invariance of the space V ( ∗ ) L p,λ ( R n ) with respect to the Hardy-Littlewood maximal operator (cf. Theorem 3.5).The paper is organized as follows. After some preliminaries on the operators underconsideration, we give the main results in Section 3. The boundedness results on thespaces V ∞ L p,λ ( R n ) and V ( ∗ ) L p,λ ( R n ) are given in Sections 3.1 and 3.2, respectively. Sec-tion 3.3 is devoted to the study of the invariance of the smaller subspace V ( ∗ )0 , ∞ L p,λ ( R n ) .Finally, we discuss additional results in Section 4 for some operators that can be seen ashybrids of potential and Hardy operators.2. Preliminaries
We use the following notation: B ( x, r ) is the open ball in R n centered at x ∈ R n andradius r > . The (Lebesgue) measure of a measurable set E ⊆ R n is denoted by | E | and χ E denotes its characteristic function. The measure of the unit ball in R n is simplydenoted by v n . We use c as a generic positive constant, i.e., a constant whose value maychange with each appearance. The expression A . A means that A ≤ c B for someindependent constant c > , and A ≈ B means A . B . A .As usual C ∞ ( R n ) stands for the class of all complex-valued infinitely differentiablefunctions on R n with compact support, and L p ( R n ) denotes the classical Lebesgue spaceequipped with the usual norm.2.1. Some classical operators.
The following class of operators was introduced in [32].
Definition 2.1.
Let < p < ∞ . A sublinear operator T is called p -admissible singulartype operator if it is bounded in L p ( R n ) and it satisfies a “size condition” of the form χ B ( x,r ) ( z ) (cid:12)(cid:12) T (cid:0) f χ R n \ B ( x, r ) (cid:1) ( z ) (cid:12)(cid:12) . χ B ( x,r ) ( z ) Z R n \ B ( x, r ) | f ( y ) || y − z | n dy for every x ∈ R n and r > . An example of p -admissible singular type operators is the Hardy-Littlewood maximaloperator
M f ( x ) := sup t> | B ( x, t ) | Z B ( x,t ) | f ( y ) | dy, x ∈ R n . It is well know that the maximal operator controls various other important operatorsof harmonic analysis. This is the case of the sharp maximal function(2.1) M ♯ f ( x ) := sup t> | B ( x, t ) | Z B ( x,t ) | f ( y ) − f B ( x,r ) | dy, with f B = | B | R B f ( z ) dz . By straightforward calculations, we have(2.2) ( M ♯ f )( x ) ≤ M f )( x ) , x ∈ R n . The class above includes also singular integral operators S , defined by(2.3) Sf ( x ) := Z R n K ( x, y ) f ( y ) dy := lim ε → Z | x − y | >ε K ( x, y ) f ( y ) dy, which are bounded in L p ( R n ) and whose kernel satisfies(2.4) | K ( x, y ) | . | x − y | − n , for all x = y. This is the case of Calderón-Zygmund operators with “standard kernels” (cf. [13, p. 99]).
A. Ç. ALABALIK, A. ALMEIDA, AND S. SAMKO
Other examples of p -admissible singular type operators are the multidimensional Hardyoperators H and H , defined by Hf ( x ) := 1 | x | n Z | y | < | x | f ( y ) dy and H f ( x ) := Z | y | > | x | f ( y ) | y | n dy. Using that | x − y | < | x | in the integral defining H , we get the pointwise estimate(2.5) H (cid:0) | f | (cid:1) ( x ) ≤ n v n M f ( x ) , x ∈ R n . We shall consider more general Hardy type operators, H α and H α , ≤ α < n , definedfor appropriate functions f by H α f ( x ) := | x | α − n Z | y | < | x | f ( y ) dy and H α f ( x ) := | x | α Z | y | > | x | f ( y ) | y | n dy. It can be easily shown that the operator H α is now dominated by the fractional maximaloperator M α f ( x ) := sup t> | B ( x, t ) | − αn Z B ( x,t ) | f ( y ) | dy, x ∈ R n , which in turn can be estimated by the Riesz potential operator I α f ( x ) := Z R n f ( y ) | x − y | n − α dy, x ∈ R n . More precisely, for < α < n there holds(2.6) (cid:12)(cid:12) H α f ( x ) (cid:12)(cid:12) ≤ v n n − α (cid:0) M α f (cid:1) ( x ) ≤ n − α I α (cid:0) | f | (cid:1) ( x ) , x ∈ R n . The Hardy operator H α is also dominated by the Riesz potential operator. The point-wise estimate below should be known, but since we did not find a reference in the literaturewe take the opportunity to give a simple proof of it. Lemma 2.2. If < α < n , then we have (2.7) (cid:12)(cid:12) H α f ( x ) (cid:12)(cid:12) ≤ n − α I α (cid:0) | f | (cid:1) ( x ) , x ∈ R n . Proof.
The inequality follows from the estimate | x | α | y | n ≤ n − α | x − y | n − α , for | y | > | x | . Putting t = | x || y | and x ′ = x | x | , y ′ = y | y | , the latter is equivalent to t α ≤ n − α (cid:16) | y || x − y | (cid:17) n − α or t α ≤ n − α | tx ′ − y ′ | n − α which is a consequence of having t < and | tx ′ − y ′ | ≤ . (cid:3) Pointwise estimates for modulars.
The estimates given in the next two lemmasare taken from [32, Theorems 4.1 and 4.3].
Lemma 2.3.
Let < p < ∞ and ≤ λ < n . If T is a p -admissible sublinear singulartype operator, then M p,λ ( T f ; x, r ) . r n − λ (cid:16) Z ∞ r t λ − np − (cid:0) M p,λ ( f ; x, t ) (cid:1) p dt (cid:17) p with the implicit constant independent of x ∈ R n , r > and f ∈ L ploc ( R n ) . NVARIANCE OF VANISHING MORREY SUBSPACES 5
Although the previous lemma is formulated for functions f in L ploc ( R n ) , the finitenessof the right-hand side implies that the function f must have already some prescribedbehaviour at infinity which, together with the size condition, ensures the well-posednessof the operator T . Remark . The estimate given in Lemma 2.3 implies that p -admissible sublinear singulartype operators are bounded on L p,λ ( R n ) and also on the vanishing space V L p,λ ( R n ) , with < p < ∞ and ≤ λ < n (cf. [32]). In particular, all the operators M , M ♯ , S , H and H are bounded on both spaces L p,λ ( R n ) and V L p,λ ( R n ) . Lemma 2.5.
Let < α < n , < p < n/α , /q = 1 /p − α/n and ≤ λ, µ < n . Then M q,µ ( I α f ; x, r ) . r n − µ (cid:16) Z ∞ r t λp − nq − (cid:0) M p,λ ( f ; x, t ) (cid:1) p dt (cid:17) q with the implicit constant not depending on x ∈ R n , r > and f ∈ L ploc ( R n ) .Remark . The estimate given in Lemma 2.5 implies a Sobolev-Spanne result on the L p,λ → L q,µ - boundedness of I α , under the additional assumptions ≤ λ < n − αp and λ/p = µ/q . Moreover, the same estimate was used in [32] to show that the Rieszpotential operator I α , and consequently the fractional maximal operator M α , are alsobounded from the the vanishing space V L p,λ ( R n ) into the vanishing space V L q,µ ( R n ) .3. Main results
In this section we show that both subspaces V ∞ L p,λ ( R n ) and V ( ∗ ) L p,λ ( R n ) are invari-ant with respect to the operators mentioned above. Since the the boundedness of suchoperators is already known in the whole Morrey space, we only have to show that thecorresponding vanishing properties are preserved under the action of those operators.3.1. Preservation of the property ( V ∞ ) .Theorem 3.1. Let < p < ∞ and ≤ λ < n . Then any p -admissible sublinear singulartype operator T is bounded in V ∞ L p,λ ( R n ) .Proof. The proof is based on the modular estimate given in Lemma 2.3. Let f ∈ V ∞ L p,λ ( R n ) . For any ε > there exists R = R ( ε ) > such that M p,λ ( f ; x, t ) < ε for every t ≥ R and all x ∈ R n . Thus for r ≥ R we have M p,λ ( T f ; x, r ) . ε r n − λ (cid:16) Z ∞ r t λ − np − dt (cid:17) p . ε with the implicit constants independent of x and r . This shows that lim r →∞ sup x ∈ R n M p,λ ( T f ; x, r ) = 0 and hence T f ∈ V ∞ L p,λ ( R n ) . (cid:3) Corollary 3.2. If < p < ∞ and ≤ λ < n , then the operators M , M ♯ , H , H and S are bounded in V ∞ L p,λ ( R n ) . In the sequel T α , < α < n , stands for any of the operators I α , M α , H α and H α above. Theorem 3.3.
Let < α < n , < p < n/α , ≤ λ < n − αp , /q = 1 /p − α/n and λ/p = µ/q . Then T α is bounded from V ∞ L p,λ ( R n ) into V ∞ L q,µ ( R n ) . A. Ç. ALABALIK, A. ALMEIDA, AND S. SAMKO
Proof.
Since I α is bounded from L p,λ ( R n ) into L q,µ ( R n ) (this is a well known result bySpanne and published in [23]), the same norm inequalities also hold for the operators M α , H α and H α in virtue of the estimates (2.6) and (2.7). Moreover, by the sameestimates, the preservation of the vanishing property at infinity by the maximal and theHardy operators follows from the corresponding preservation by the action of the Rieszpotential operator. Hence, it remains to show that sup x ∈ R n M p,λ ( f ; x, r ) → ⇒ sup x ∈ R n M q,µ ( I α f ; x, r ) → as r → ∞ . This can be done as in the proof of Theorem 3.1, but now using the modular estimategiven in Lemma 2.5 instead of that in Lemma 2.3. Nevertheless, we write the proof inslightly different terms by applying the Lebesgue dominated convergence theorem. Forany x ∈ R n and r > , we have M q,µ ( I α f ; x, r ) . r n − µ (cid:16) Z ∞ r t λp − nq − (cid:0) M p,λ ( f ; x, t ) (cid:1) p dt (cid:17) q = (cid:16) Z ∞ s λp − nq − (cid:0) M p,λ ( f ; x, rs ) (cid:1) p ds (cid:17) q . (3.1)Since the implicit constant does not depend on x ∈ R n and r > , and the integrandhas admits an integrable dominant (note that f ∈ L p,λ ( R n ) and λ/p < n/q ), then theright-hand side of (3.1) tends to as r → ∞ (uniformly on x ). This gives the desiredresult. (cid:3) Theorem 3.4.
Let < α < n , ≤ λ < n , < p < ( n − λ ) /α and /q = 1 /p − α/ ( n − λ ) .Then T α is bounded from V ∞ L p,λ ( R n ) into V ∞ L q,λ ( R n ) .Proof. As in the proof of Theorem 3.3 we only have to prove the statement for the Rieszpotential operator. The L p,λ → L q,λ - boundedness of I α is well known (cf. [1]). Toshow the preservation of the vanishing property at infinity we make use of the pointwiseestimate of the Riesz potential operator in terms of the maximal function. It is knownthat there exists c > such that(3.2) | I α f ( x ) | ≤ c (cid:0) M f ( x ) (cid:1) pq k f k − pq p,λ for all f ∈ L p,λ ( R n ) and x ∈ R n (see, for instance, [9], [2, Chapter 7]). From (3.2) we get M q,λ (cid:0) I α f ; x, r (cid:1) . k f k q − pp,λ M p,λ (cid:0) M f ; x, r (cid:1) for r > and x ∈ R n . If f ∈ V ∞ L p,λ ( R n ) then M f ∈ V ∞ L p,λ ( R n ) by Corollary 3.2.Consequently, we have I α f ∈ V ∞ L q,λ ( R n ) taking into account the previous estimate. (cid:3) Preservation of the property ( V ∗ ) . First we show that the space V ( ∗ ) L p,λ ( R n ) is invariant under the action of the Hardy-Littlewood maximal operator. Theorem 3.5.
Let < p < ∞ and ≤ λ < n . Then the maximal operator M is boundedin V ( ∗ ) L p,λ ( R n ) .Proof. Since M is bounded in L p,λ ( R n ) (cf. [11]) we only have to show that it preservesthe vanishing property ( V ∗ ), that is lim N →∞ A N,p ( f ) = 0 ⇒ lim N →∞ A N,p ( M f ) = 0 . Given x ∈ R n and N ∈ N , we split f into f = f + f , with f := f χ Ω x,N/ , f = f χ R n \ Ω x,N/ , NVARIANCE OF VANISHING MORREY SUBSPACES 7 where, for short, we use the notation Ω x,N := B ( x, ∩ (cid:0) R n \ B (0 , N ) (cid:1) . Since M is sublinear, we have(3.3) A N,p ( M f ) . A N,p ( M f ) + A N,p ( M f ) . We show next that both quantities in the the right-hand side of (3.3) tend to zero as N → ∞ . The boundedness of M in L p ( R n ) gives Z B ( x, (cid:0) M ( f )( y ) (cid:1) p χ N ( y ) dy ≤ Z R n (cid:0) M ( f )( y ) (cid:1) p dy . Z R n | f ( y ) | p dy = Z Ω x,N/ | f ( y ) | p dy (3.4)with the implicit constant independent of x , N and f . Since f ∈ V ( ∗ ) L p,λ ( R n ) , the righthand side above tends to zero uniformly on x as N → ∞ (note that the property ( V ∗ ) does not depend on the particular value of the radius taken in the balls centered at x , cf.[5, Lemma 3.4]). Therefore, lim N →∞ A N,p ( M f ) = 0 . Now we deal with the second term in the sum in (3.3). Let ε > be arbitrary. Thenthere exists t > such that t λ − n < ε for all t ≥ t . For such fixed t , we have Z B ( x, (cid:0) M ( f )( y ) (cid:1) p χ N ( y ) dy . I ( x, N ) + I ( x, N ) where I ( x, N ) := Z B ( x, χ N ( y ) sup
Using this, the Minkowski’s inequality and a simple change of variables, we get I ( x, N ) ≤ Z B ( x, χ N ( y ) (cid:20)Z R n χ B (0 ,t ) ( z ) | f ( y − z ) | dz (cid:21) p dy ≤ Z R n χ B (0 ,t ) ( z ) (cid:20) sup v ∈ R n Z B ( v, | f ( u ) | p χ N −| z | ( u ) du (cid:21) /p dz ! p =: (cid:18)Z R n g N ( z ) dz (cid:19) p with the interpretation χ a := 1 if a ≤ and χ a := χ R n \ B (0 ,a ) if a > . This gives an uniform bound for I ( x, N ) . Since f ∈ L p,λ ( R n ) and g N ( z ) has an integrablemajorant (depending on t ), an application of the Lebesgue convergence theorem showsthat R R n g N ( z ) dz → as N → ∞ , which implies that I ( x, N ) → uniformly on x , as N → ∞ . The proof is complete. (cid:3)
Theorem 3.6.
Let < p < ∞ and ≤ λ < n . Then the operators M ♯ , H and H arebounded in V ( ∗ ) L p,λ ( R n ) .Proof. The boundedness of M ♯ and H in V ( ∗ ) L p,λ ( R n ) is a consequence of Theorem 3.5and inequalities (2.2), (2.5). The case of the Hardy operator H requires a differentapproach since it can not be estimated by the maximal function. Observing that lim | z |→∞ H f ( z ) = 0 , then for any ε > there exists N ε ∈ N such that |H f ( y ) | ≤ ( ε/v n ) /p for all | y | ≥ N ε .Therefore Z R n (cid:12)(cid:12) H f ( y ) (cid:12)(cid:12) p χ B ( x, ∩{ y : | y | >N } ( y ) dy ≤ ε/v n Z R n χ B ( x, ∩{ y : | y | >N } ( y ) dy ≤ ε for all N ≥ N ε , uniformly on x ∈ R n . This shows that A N,p (cid:0) H f (cid:1) → as N → ∞ . (cid:3) Remark . As regards the preservation of the property ( V ∗ ) by the action of the Hardyoperator H , one can formulate a stronger result. Indeed, as we can see from the proofabove, H is bounded from the whole space Morrey L p,λ ( R n ) into V ( ∗ ) L p,λ ( R n ) .As in Section 3.1 suppose again that T α , < α < n , denotes any of the operators I α , M α , H α and H α . Theorem 3.8.
Let < α < n , ≤ λ < n , < p < ( n − λ ) /α and /q = 1 /p − α/ ( n − λ ) .Then T α is bounded from V ( ∗ ) L p,λ ( R n ) into V ( ∗ ) L q,λ ( R n ) .Proof. By (3.2) we get A N,q (cid:0) I α f (cid:1) . k f k q − pp,λ A N,p ( M f ) with the implicit constant independent of f and N ∈ N . If f ∈ V ( ∗ ) L p,λ ( R n ) then M f ∈ V ( ∗ ) L p,λ ( R n ) by Theorem 3.5. Consequently, we also have I α f ∈ V ( ∗ ) L q,λ ( R n ) bythe previous estimate. (cid:3) NVARIANCE OF VANISHING MORREY SUBSPACES 9
Invariance of the closure of C ∞ ( R n ) . As shown in [5], the subspace V ( ∗ )0 , ∞ L p,λ ( R n ) := V L p,λ ( R n ) ∩ V ∞ L p,λ ( R n ) ∩ V ( ∗ ) L p,λ ( R n ) coincides with the closure of the class C ∞ ( R n ) in Morrey norm. Note that this closureplays an important role in harmonic analysis on Morrey spaces since its dual provides apredual space for Morrey spaces (cf. [2, 3, 31, 41]). Moreover (cf. [5]), we have the strictembeddings V ( ∗ )0 , ∞ L p,λ ( R n ) $ V L p,λ ( R n ) ∩ V ∞ L p,λ ( R n ) $ V L p,λ ( R n ) $ L p,λ ( R n ) . The next corollaries are immediate consequences of the results obtained in Sections 3.1,3.2 and the already known corresponding boundedness in the vanishing space V L p,λ ( R n ) . Corollary 3.9.
Let < p < ∞ and ≤ λ < n . Then the operators M , M ♯ , H and H are bounded in V ( ∗ )0 , ∞ L p,λ ( R n ) . Corollary 3.10.
Let < α < n , ≤ λ < n , < p < ( n − λ ) /α and /q = 1 /p − α/ ( n − λ ) .Then T α is bounded from V ( ∗ )0 , ∞ L p,λ ( R n ) into V ( ∗ )0 , ∞ L q,λ ( R n ) . We end this section with a further result for singular integral operators S defined by(2.3), with the kernel satisfying (2.4). One knows that S is bounded in V L p,λ ( R n ) , for < p < ∞ and ≤ λ < n (cf. [32, Theorem 5.1]). On the other hand, we have seenthat S is also bounded in V ∞ L p,λ ( R n ) (cf. Corollary 3.2). Unfortunately, we do not knowwhether S preserves the vanishing property ( V ∗ ) . Nevertheless, we have the followingresult: Theorem 3.11.
Let < p < ∞ and ≤ λ < n . Then S is bounded from V ( ∗ )0 , ∞ L p,λ ( R n ) into itself.Proof. For bounded compactly supported functions f , there holds(3.5) | Sf ( y ) | ≤ c | y | n for sufficiently large | y | , with c > not depending on y . In fact, suppose that supp f ⊂ B (0 , M ) . We have | y − z | ≥ | y | − M ≥ | y | for | z | ≤ M and | y | ≥ M. Hence (3.5) follows thanks to the size condition (2.4).Let now f ∈ V ( ∗ )0 , ∞ L p,λ ( R n ) . Then there exists a sequence ( f k ) ⊂ C ∞ ( R n ) such that f k → f in L p,λ ( R n ) as k → ∞ . By the continuity of S in L p,λ ( R n ) , we get Sf = S ( lim k →∞ f k ) = lim k →∞ ( Sf k ) . Thus we have Sf ∈ V ( ∗ )0 , ∞ L p,λ ( R n ) since this subspace is closed in L p,λ ( R n ) . (cid:3) Additional results
In this section we consider the following hybrids of Hardy and potential operators: K β f ( x ) := 1 | x | β Z | y | < | x | f ( y ) | x − y | n − β dy and K β f ( x ) := Z | y | > | x | f ( y ) | y | β | x − y | n − β dy (with < β ≤ n ). As observed in [26], K β and K β are integral operators bounded in L p ( R n ) , < p < ∞ , with the corresponding kernels satisfying the size condition (2.4).Consequently, by Theorem 3.1 we have the following result: Theorem 4.1.
Let < p < ∞ , ≤ λ < n and < β ≤ n . Then both operators K β and K β are bounded on V ∞ L p,λ ( R n ) . Acknowledgments.
A. Almeida was supported by CIDMA (Center for Researchand Development in Mathematics and Applications) and FCT (Portuguese Foundationfor Science and Technology) within project UID/MAT/04106/2019. The research of S.Samko was supported by Russian Foundation for Basic Research under the grants 19-01-00223 and 18-01-00094-a.
References
1. D.R. Adams:
A note on Riesz potentials , Duke. Math. J. (1975), 765–778.2. D.R. Adams: Morrey Spaces , Lecture Notes in Applied and Numerical Harmonic Analysis,Birkhäuser, 2015.3. D.R. Adams, J. Xiao:
Morrey spaces in harmonic analysis , Ark. Mat. (2012), 201–230.4. A. Akbulut, V.S. Guliyev, R. Mustafayev: On the boundedness of the maximal operator and singularintegral operators in generalized Morrey spaces , Math. Bohemica (2012), 27–43.5. A. Almeida, S. Samko:
Approximation in Morrey spaces , J. Funct. Anal. (2017), 2392–2411.6. A. Almeida, S. Samko:
Approximation in generalized Morrey spaces , Georgian Math. J. (2)(2018), 155–168.7. V.I. Burenkov, A. Gogatishvili, V.S. Guliyev, R. Mustafayev:
Boundedness of the fractional maximaloperator in Morrey-type spaces , Complex Var. Elliptic Equ.. (2010), 739–758.8. V.I. Burenkov, P. Jain, T.V. Tararykova: On boundedness of the Hardy operator in Morrey-typespaces , Eurasian Math. J. (2011), 52–80.9. E. Burtseva, N. Samko: Weighted Adams type theorem for the Riesz fractional integral in generalizedMorrey spaces , Fract. Calc. Appl. Anal. (2016), 954–972.10. S. Campanato: Proprietà di una famiglia di spazi funzioni , Ann. Scuola Norm. Sup. Pisa (1964),137–160.11. F. Chiarenza, M. Frasca: Morrey spaces and Hardy-Littlewood maximal function , Rend. Math. (1987) 273–279.12. F. Chiarenza, M. Franciosi: A generalization of a theorem by C. Miranda , Ann. Mat. Pura Appl.,IV. Ser. 161, (1992) 285–297.13. J. Duoandikoetxea:
Fourier Analysis , Amer. Math. Soc., “Graduate Studies”, vol. 29, 2001.14. M. Giaquinta:
Multiple integrals in the calculus of variations and non-linear elliptic systems , Prince-ton Univ. Press, Princeton, 1983.15. V.S. Guliyev:
Boundedness of the maximal, potential and singular operators in the generalized Morreyspaces , J. Inequal. Appl., Art. ID503948 (2009), 20pp.16. T. Kato:
Strong solutions of the Navier-Stokes equation in Morrey spaces , Bol. Soc. Bras. Mat. (1992), 127–155.17. K. Kurata, S. Nishigaki, S. Sugano: Boundedness of integral operators on generalized Morrey spacesand its application to Schrödinger operators , Proc. Amer. Math. Soc. (2000), 1125–1134.18. P.G. Lemarié-Rieusset:
The Navier-Stokes problem in the 21st century , CRC Press, Boca Raton, FL,2016.19. D. Lukkassen, A. Medell, L.-E. Persson, N. Samko:
Hardy and singular operators in weighted gen-eralized Morrey spaces with applications to singular integral equations , Math. Methods Appl. Sci. (11) (2012), 1300–1311.20. T. Mizuhara: Boundedness of some classical operators on generalized Morrey spaces , ICM-90 SatelliteConf. Proc., S. Igari (Ed.) (1991), pp. 183–189.
NVARIANCE OF VANISHING MORREY SUBSPACES 11
21. C.B. Morrey:
On the solutions of quasi-linear elliptic partial differential equations , Trans. Amer.Math. Soc. (1938), 126–166.22. E. Nakai: Hardy-Littlewood maximal operator, singular integral operators and Riesz potentials ongeneralized Morrey spaces , Math. Nachr. (1994), 95–103.23. J. Peetre:
On the theory of L p,λ spaces , J. Funct. Anal. (1969), 71–87.24. L.E. Persson, N. Samko: Weighted Hardy and potential operators in the generalized Morrey spaces ,J. Math. Anal. Appl. (2011), 792–806.25. L.E. Persson, M.A. Ragusa, N. Samko, P. Wall:
Commutators of Hardy operators in vanishingMorrey spaces , AIP Conf. Proc. 1493 (2012), 859–866.26. L.E. Persson, N. Samko, P. Wall:
Calderón-Zygmund type operators in weighted generalized Morreyspaces , J. Fourier Anal. Appl. (2016), 413–426.27. L. Pick, A. Kufner, O. John, S. Fučík: Function spaces , Vol. 1, 2nd. ed., De Gruyter Series inNonlinear Analysis and Applications 14, Berlin, 2013.28. H. Rafeiro, N. Samko, S. Samko:
Morrey-Campanato spaces: an overview , In: Karlovich, Yi., Rodino,L., Silbermann, B., Spitkovsky, IM. (eds.)
Operator Theory, Pseudo-Differential Equations and Math-ematical Physics, Advances and Applications , vol. 228, pp. 293–323, Springer, Basel, 2013.29. M.A. Ragusa:
Commutators of fractional integral operators on vanishing Morrey spaces , J. GlobalOptim. (2008) 361–368.30. M. Rosenthal, H. Triebel: Calderón-Zygmund operators in Morrey spaces , Rev. Mat. Complut. (2014), 1–11.31. M. Rosenthal, H. Triebel: Morrey spaces, their duals and preduals , Rev. Mat. Complut. (2015),1–30.32. N. Samko: Maximal, potential and singular operators in vanishing generalized Morrey spaces , J.Glob. Optim. (2013), 1385–1399.33. N. Samko: Weighted Hardy operators in the local generalized vanishing Morrey spaces , Positivity (2013), 683–706.34. Y. Sawano, S. Sugano, H. Tanaka: A note on generalized fractional integral operators on generalizedMorrey spaces , Bound Value Probl. INDO. 835865 (2009)35. S. Shirai:
Necessary and sufficient conditions for boundedness of commutators of fractional integraloperators on classical Morrey spaces , Hokkaido Math. J. (3) (2006), 683–696.36. L.G. Softova: Singular integrals and commutators in generalized Morrey spaces , Acta Math. Sin.(Engl. Ser.) (2006), 757–766.37. L.G. Softova: Morrey-type regularity of solution to parabolic problems with discontinuous data ,Manuscr. Math. (2011), 365–382.38. S. Sugano, H. Tanaka:
Boundedness of fractional integral operators on generalized Morrey spaces ,Sci. Math. Jpn. (3) (2003), 531–540.39. M.E. Taylor: Tools for PDE: Pseudodifferential Operators, Paradifferential Operators and LayerPotentials , Math. Surveys and Monogr., Vol. 81, AMS, Providence, 2000.40. H. Triebel:
Local function spaces, Heat and Navier-Stokes equations , EMS Tracts in Mathematics20, 2013.41. H. Triebel:
Hybrid function spaces, Heat and Navier-Stokes equations , EMS Tracts in Mathematics24, 2015.42. C. Vitanza:
Functions with vanishing Morrey norm and elliptic partial differential equations . In:Proceedings of Methods of Real Analysis and Partial Differential Equations, Capri, pp. 147–150,Springer, 1990.43. C. Vitanza:
Regularity results for a class of elliptic equations with coefficients in Morrey spaces . Ric.Mat. , No.2 (1993), 265–281 .44. W. Yuan, W. Sickel, D. Yang: Calderón’s first and second complex interpolations of closed subspacesof Morrey spaces , Sci. China Math. (2015), 1835–1908.45. C. Zorko: Morrey spaces , Proc. Amer. Math. Soc. (1986), 586–592. Department of Mathematics, Institute of Natural and Applied Sciences, Dicle Uni-versity, 21280 Diyarbakir, Turkey
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