L^p-L^q boundedness of (k, a)-Fourier multipliers with applications to Nonlinear equations
aa r X i v : . [ m a t h . F A ] J a n L p - L q BOUNDEDNESS OF ( k, a ) -FOURIER MULTIPLIERS WITHAPPLICATIONS TO NONLINEAR EQUATIONS VISHVESH KUMAR AND MICHAEL RUZHANSKY
Abstract.
The ( k, a )-generalised Fourier transform is the unitary operator definedusing the a -deformed Dunkl harmonic oscillator. The main aim of this paper is toprove L p - L q boundedness of ( k, a )-generalised Fourier multipliers. To show the bounded-ness we first establish Paley inequality and Hausdorff-Young-Paley inequality for ( k, a )-generalised Fourier transform. We also demonstrate applications of obtained results tostudy the well-posedness of nonlinear partial differential equations. Introduction and Basics on ( k, a ) -generalised Fourier transform In his seminal paper [28], H¨ormander initiated the study of boundededness of the trans-lation invariant operators on R N . The translation operators R N can be characterised usingthe classical Fourier transform on R N and therefore they are also known as Fourier mul-tipliers. The boundedness of Fourier multipliers is useful to solve problems in the area ofmathematical analysis, in particular, in PDEs. H¨ormander established the L p - bounded-ness and L p - L q boundedness of Fourier multipliers on R N . After that, L p -boundedness ofFourier multipliers has been investigated by several researchers in many different setting,we cite here [28, 5, 16, 37, 12, 13, 23, 38, 41, 27, 24] to mention a few of them. In particular, L p -boundedness of multipliers was established in [38] for the one dimensional Dunkl trans-form and very recently in [24] in the multidimensional setting. Recently, the researchershave turned their attention to establish the boundedness of L p - L q multipliers for therange 1 < p ≤ ≤ q < ∞ , see [1, 3, 4, 14, 15, 18, 34]. Precisely, the second author andhis collaborators started investigating the H¨ormander L p - L q Fourier multipliers theoremand its different consequences for locally compact groups and on homogeneous manifolds.Such analysis includes the Hardy-Littlewood inequality, spectral multipliers theorems and
Mathematics Subject Classification.
Primary 42B10; 42B37 Secondary 42B15; 33C45.
Key words and phrases. ( k, a )-generalised Fourier transform; a -deformed Dunkl oscillator; ( k, a )-Fourier multipliers; Paley inequality; Hausdorff-Young-Paley inequality; Nonlinear PDEs. applications to PDEs [1, 3, 2, 34]. In [15], similar results have been proved for the eigen-function expansions of anharmonic oscillators and extended to the more general settingof bi-orthogonal expansions in [14]. Ben Sa¨ıd et al. [7, 8] introduced ( k, a )-generalisedFourier transform. It generalises many important integral transforms including Fouriertransform and Dunkl transform on the Euclidean spaces R N , [9, 8]. Recently, there is agrowing interest to develop the analysis related to the ( k, a )-generalised Fourier trans-form. Notably, the uncertainty principals and Pitt inequalities [25, 30], maximal functionand translation operator [10], wavelets multipliers [35] and Hardy inequality [40] wereexplored by many researchers. In this paper, we establish L p - L q boundedness of ( k, a )-Fourier multipliers theorem using the ( k, a )-generalised Fourier transform. The proof themain result hinges upon the Paley inequality and Hausdorff-Young-Palay inequality for( k, a )-generalised Fourier transform obtained by using the Hausdorff-Young inequalityestablished in [30, 25].To describe our main result let us recall the classical H¨ormander Fourier multiplierstheorem settled in [28]: For 1 < p ≤ ≤ q < ∞ , the Fourier multiplier T m : S ( R N ) →S ′ ( R N ) associated with symbol m : R N → C defined by F ( T m f )( ξ ) = m ( ξ ) F ( f )( ξ ) for ξ ∈ R N , has a bounded extension from L p ( R N ) to L q ( R N ) provided that the symbolsatisfies the condition |{ ξ ∈ R N : | m ( ξ ) | ≥ s }| ≤ s b for all s > , (1)where b = p − q , and F denotes the Euclidean Fourier transform of f defined as F ( f )( ξ ) := (2 π ) − N Z R N f ( x ) e − i h x,ξ i dx, ξ ∈ R N . Here h x, ξ i denotes the standard Euclidean inner product of two vectors x and ξ in R N and k x k will denote the Euclidean norm on R N . The Euclidean Fourier transform F on R N can be described using the spectrum information of the harmonic oscillator ∆ R N − k x k , where ∆ R N is the Laplacian on R N . In fact, Howe [29] found the following description ofthe Euclidean Fourier transform F : F := exp (cid:16) iπN (cid:17) exp (cid:16) iπ R N − k x k ) (cid:17) . (2) OUNDEDNESS OF ( k, a )-FOURIER MULTIPLIERS 3
This description has been proved to be useful to define generalisations of the Fouriertransform such as Clifford algebra-valued Fourier transform and fractional Fourier trans-form. These constructions have been explained in an excellent overview article [21]. Onthe other hand, Dunkl [19, 20] presented a generalisation of the Euclidean Fourier trans-form and Euclidean Laplacian on R N , which is now known as Dunkl transform (see [22])and Dunkl Laplacian, and are usually denoted by F k and ∆ k , respectively, using theroot system R ⊂ R N , a reflection group G ⊂ O ( N, R ) generated by the root reflections r α , α ∈ R , and a multiplicity function k : R → R + such that k is G invariant. Weset k ( α ) = k α , h k i = P α ∈R k α , v k ( x ) = Q α ∈R |h α, x i| k α , v k,a ( x ) := k x k − a v k ( x ) . Define L pk,a ( R N ) := L p ( R N , v k,a dx ) and dµ k,a ( x ) = v k,a dx. To describe the Dunkl Laplacian, let us define the first order Dunkl operator for ξ ∈ R N and a fixed multiplicity function k by T ξ ( k ) f ( x ) = ∂ ξ f ( x ) + X α ∈R + k α h α, ξ i f ( x ) − f ( r α x ) h α, x i , f ∈ C ( R N ) , where ∂ ξ is the direction derivation in the direction of ξ and R + denotes the positive rootsubsystem. Let us fix an orthonormal basis { ξ , ξ , . . . , ξ N } for the inner product space( R N , h· , ·i ) and write T ξ j ( k ) as T j ( k ) for j ∈ { , , . . . , N } . Then the Dunkl Laplacian isdefined by ∆ k = P Nj =1 T j ( k ) . The Dunkl Laplacian has explicit form and also plays a veryimportant role in the Dunkl analysis (see [6, 36] for more details and related analysis).When the multiplicity function is trivial (i.e., k ≡
0) then F k and ∆ k turn out be to justthe Euclidean Fourier transform F and the Euclidean Laplacian ∆ R N , respectively. Usingthe Dunkl Laplacian one can define the Dunkl harmonic oscillator (or Dunkl-Hermiteoperator) as ∆ k − k x k . Ben Sa¨ıd et al. [7] considered the a -deformed Dunkl harmonicoscillator given by ∆ k,a := k x k − a ∆ k − k x k a , a > . By making use of this a -deformed Dunkl harmonic oscillator ∆ k,a , they introduced a twoparameters unitary operator, ( k, a ) -generalised Fourier transform , F k,a on L k,a ( R N ) , by F k,a := exp (cid:20) iπ (cid:18) a (2 h k i + n + a − (cid:19)(cid:21) exp (cid:20) iπ a (∆ k,a ) (cid:21) . (3)The ( k, a )-generalised Fourier transform F k,a includes some prominent transforms onthe Euclidean space R N : VISHVESH KUMAR AND MICHAEL RUZHANSKY • For a = 2 and k > F k,a is the Dunkl transform [22]. • For a = 2 and k ≡ , F k,a is the Euclidean Fourier transform [29]. • For a = 1 and k ≡ , F k,a is the Hankel transform appearing as the unitaryinversion operator of the Schr¨odinger model of the minimal representation of thegroup O ( N + 1 ,
2) (see [31, 32, 33]).For a > a + 2 h k i + N − > , the ( k, a )-generalised Fourier transform F k,a is abijective linear operator such that kF k,a ( f ) k L k,a ( R N ) = k f k L k,a ( R N ) . (4)By the Schwartz kernel theorem there exists a distribution kernel B k ( ξ, x ) such that F k,a f ( ξ ) = c k,a Z R N B k,a ( ξ, x ) f ( x ) dµ k,a ( x )with a symmetric kernel B k,a ( ξ, x ) ([7]).The next lemma presents some conditions on N, k , and a such that kernel B k,a ( ξ, x ) isuniformly bounded. Lemma 1.1. [30, Lemma 2.8]
Assume N ≥ , k ≥ , a + 2 h k i + N − > , and thatexactly one of the following additional assumption holds: (i) N = 1 and a > ; (ii) a ∈ { , } ;(iii) k = 0 and a = m for some m ∈ N . Then B k,a is uniformly bounded, that is, | B k,a ( ξ, x ) | ≤ M for all x, ξ ∈ R N , where M is afinite constant that depends only on N, k, and a. Theorem 1.2.
Assume that
N, k, and a satisfy the assumption of Lemma 1.1. For ≤ p ≤ , fix p ′ = pp − . Then for f ∈ L pk,a ( R N ) we have kF k,a f k L p ′ k,a ( R N ) ≤ C k f k L pk,a ( R N ) , (5) where C = M /p − . It was conjectured by Gorbachev et al. [25] that if a + 2 h k i + N − > | B k,a ( ξ, x ) | ≤ B k,a (0 , x ) = 1 for all x, ξ ∈ R N . So in this case, the constant C inHausdorff-Young inequality (5) becomes 1 . OUNDEDNESS OF ( k, a )-FOURIER MULTIPLIERS 5
From this point onward, we always assume that
N, k and a either satisfy assumptionsof Lemma 1.1 with N ≥ , k ≥ a + 2 h k i + N − > , or, a + 2 h k i + N − > k, a )-generalised Fourier transform we are now in aposition to state our results. The main result of this paper is the following theorem L p - L q boundedness of ( k, a )-Fourier multipliers A for the range 1 < p ≤ ≤ q < ∞ . Indeed, wehave k A k L pk,a ( R N ) → L qk,a ( R N ) . sup s> s (cid:20)Z { ξ ∈ R N : | h ( ξ ) |≥ s } dµ k,a ( ξ ) (cid:21) p − q , where h is the symbol of the ( k, a )-Fourier multiplier A , this means that, F k,a ( Af )( ξ ) = h ( ξ ) F k,a f ( ξ ) for ξ ∈ R N and for f in a suitable function space. The main tool to establishthis result is the following Hausdorff-Young Paley inequality for ( k, a )-generalised Fouriertransform: For 1 < p ≤ , < p ≤ b ≤ p ′ ≤ ∞ , where p ′ = pp − and for a positive function ψ defined on R N we have (cid:18)Z R N (cid:16) |F k,a f ( ξ ) | ψ ( ξ ) b − p ′ (cid:17) b dµ k,a ( ξ ) (cid:19) b . (cid:16) sup t> t Z ξ ∈ R Nψ ( ξ ) ≥ t dµ k,a ( ξ ) (cid:17) b − p ′ k f k L pk,a ( R N ) . (6)Next, we will present applications of our main results in the context of well-posedness ofnonlinear abstract Cauchy problems in the space L ∞ (0 , T, L k,a ( R N )) . First, we considerthe heat equation u t − | Bu ( t ) | p = 0 , u (0) = u , (7)where B is a linear operator on L k,a ( R N ) and 1 < p < ∞ . We study local well-posednessof the above heat equation (7). Secondly, we consider the initial value problem for thenonlinear wave equation u tt ( t ) − b ( t ) | Bu ( t ) | p = 0 , (8)with the initial condition u (0) = u , u t (0) = u , where b is a positive bounded functiondepending only on time, B is a linear operator in L k,a ( R N ) and 1 < p < ∞ . We explorethe global and local well-posedness of (8) under some condition on function b. We organise the paper in following way: In the next section we will state and presentthe proof of Paley inequality and Hausdorff-Young-Paley inequality. Then, we give theproof of our main result concerning the L p - L q boundedness of ( k, a )-Fourier multipliers VISHVESH KUMAR AND MICHAEL RUZHANSKY and its consequences. In the last section, the applications of the result proved in previoussection will be discussed. 2.
Main results
Throughout the paper, we shall use the notation A . B to indicate A ≤ cB for asuitable constant c >
0. In this section, we will present our main results. In the proofswe follows the ideas in the papers [2, 3]. The first result is the Paley inequality for the( k, a )-generalised Fourier transform.
Theorem 2.1.
Suppose that ψ is a positive function on R N satisfying the condition M ψ := sup t> t Z ξ ∈ R Nψ ( ξ ) ≥ t dµ k,a ( ξ ) < ∞ . (9) Then for f ∈ L pk,a ( R N ) , < p ≤ , we have (cid:18)Z R N | b f ( ξ ) | p ψ ( ξ ) − p dµ k,a ( ξ ) (cid:19) p . M − pp ψ k f k L pk,a ( R N ) . (10) Proof.
Let us consider a measure ν on R N given by ν k,a ( ξ ) = ψ ( ξ ) dµ k,a ( ξ ) . (11)We define the corresponding L p ( R N , ν k,a )- space, 1 ≤ p < ∞ , as the space of all complex-valued function f defined by R N such that k f k L p ( R N ,ν k,a ) := (cid:18)Z R N | f ( ξ ) | p ψ ( ξ ) dν k,a ( ξ ) (cid:19) p < ∞ . We define a sublinear operator T for f ∈ L pk,a ( R N ) by T f ( ξ ) = |F k,a f ( ξ ) | ψ ( ξ ) ∈ L p ( R N , ν k,a ) . We will show that T is well-defined and bounded from L pk,a ( R N ) to L p ( R N , ν k,a ) for any1 < p ≤ . In other words, we claim the following estimate: k T f k L p ( R N ,ν k,a ) = Z R N | b f ( ξ ) | p ψ ( ξ ) p ψ ( ξ ) dµ k,a ( ξ ) ! p . M − pp ψ k f k L pk,a ( R N ) , (12) OUNDEDNESS OF ( k, a )-FOURIER MULTIPLIERS 7 which will give us the required inequality (10) with M ψ := sup t> R λ ∈ R Nψ ( ξ ) ≥ t dµ k,a ( ξ ) . We willshow that T is weak-type (2 ,
2) and weak-type (1 , . More precisely, with the distributionfunction, ν k,a ( y ; T f ) = Z ξ ∈ R N |F k,af ( ξ ) | ψ ( ξ ) ≥ y ψ ( ξ ) dµ k,a ( ξ ) , where ν k,a is given by formula (11), we show that ν k,a ( y ; T f ) ≤ M k f k L k,a ( R N ) y ! with norm M = 1 , (13) ν k,a ( y ; T f ) ≤ M k f k L k,a ( R N ) y with norm M = M ψ . (14)Then the estimate (12) follows from the Marcinkiewicz interpolation Theorem. Now, toshow (13), using Plancherel identity we get y ν k,a ( y ; T f ) ≤ sup y> y ν k,a ( y ; T f ) =: k T f k L , ∞ ( R N ,ν k,a ) ≤ k T f k L ( R N ,ν k,a ) = Z R N (cid:18) |F k,a ( ξ ) | ψ ( ξ ) (cid:19) ψ ( ξ ) dµ k,a ( ξ )= Z R N |F k,a ( ξ ) | dµ k,a ( ξ ) = k f k L k,a ( R N ) . Thus, T is type (2 ,
2) with norm M ≤ . Further, we show that T is of weak type (1 , M = M ψ ; more precisely, we show that ν k,a (cid:26) ξ ∈ R N : |F k,a ( ξ ) | ψ ( ξ ) > y (cid:27) . M ψ k f k L k,a ( R N ) y . (15)The left hand side is an integral R ψ ( ξ ) dµ k,a ( ξ ) taken over all those ξ ∈ R N for which | b f ( ξ ) | ψ ( ξ ) > y. Since |F k,a f ( ξ ) | ≤ k f k L k,a ( R N ) for all ξ ∈ R N we have (cid:26) ξ ∈ R N : |F k,a f ( ξ ) | ψ ( ξ ) > y (cid:27) ⊂ ( ξ ∈ R N : k f k L k,a ( R N ) ψ ( ξ ) > y ) , for any y > ν k,a (cid:26) ξ ∈ R N : |F k,a ( ξ ) | ψ ( ξ ) > y (cid:27) ≤ ν ( ξ ∈ R N : k f k L k,a ( R N ) ψ ( ξ ) > y ) . VISHVESH KUMAR AND MICHAEL RUZHANSKY
Now by setting w := k f k L k,a ( R N ) y , we have ν k,a ( ξ ∈ R N : k f k L k,a ( R N ) ψ ( ξ ) > y ) ≤ Z ξ ∈ R N ψ ( ξ ) ≤ w ψ ( ξ ) dµ k,a ( ξ ) . (16)Now we claim that Z ξ ∈ R N ψ ( ξ ) ≤ w ψ ( ξ ) dµ k,a ( ξ ) . M ψ w. (17)Indeed, first we notice that Z ξ ∈ R N ψ ( ξ ) ≤ w ψ ( ξ ) dµ k,a ( ξ ) = Z ξ ∈ R N ψ ( ξ ) ≤ w dµ k,a ( ξ ) Z ψ ( ξ ) dτ. By interchanging the order of integration we get Z ξ ∈ R N ψ ( ξ ) ≤ w dµ k,a ( ξ ) Z ψ ( ξ ) dτ = Z w dτ Z ξ ∈ R Nτ ≤ ψ ( ξ ) ≤ w dµ k,a ( ξ ) . Further, by making substitution τ = t , it gives Z w dτ Z λ ∈ R Nτ ≤ ψ ( ξ ) ≤ w dµ k,a ( ξ ) = 2 Z w t dt Z ξ ∈ R Nt ≤ ψ ( ξ ) ≤ w dµ k,a ( ξ ) ≤ Z w t dt Z ξ ∈ R Nt ≤ ψ ( ξ ) dµ k,a ( ξ ) . Since t Z ξ ∈ R Nt ≤ ψ ( ξ ) dµ k,a ( ξ ) ≤ sup t> t Z ξ ∈ R Nt ≤ ψ ( ξ ) dµ k,a ( ξ ) = M ψ is finite by assumption M ψ < ∞ , we have2 Z w t dt Z ξ ∈ R Nt ≤ ψ ( ξ ) dµ k,a ( ξ ) . M ψ w. This establishes our claim (17) and eventually proves (15). So, we have proved (13) and(14). Then by using the Marcinkiewicz interpolation theorem with p = 1 and p = 2 and p = 1 − θ + θ we now obtain (cid:18)Z R + (cid:18) |F k,a f ( ξ ) | ψ ( ξ ) (cid:19) p ψ ( ξ ) dµ k,a ( ξ ) (cid:19) p = k T f k L p ( R N , ν k,a ) . M − pp ψ k f k L pk,a ( R N ) . This completes the proof of the theorem. (cid:3)
Now, we record the following interpolation theorem from [11] for further use.
OUNDEDNESS OF ( k, a )-FOURIER MULTIPLIERS 9
Theorem 2.2.
Let dµ ( x ) = ω ( x ) dµ ′ ( x ) , dµ ( x ) = ω ( x ) dµ ′ ( x ) . Suppose that
Theorem 2.3.
Let < p ≤ , and let < p ≤ b ≤ p ′ ≤ ∞ , where p ′ = pp − . If ψ is apositive function on R N such that M ψ := sup t> t Z λ ∈ R Nψ ( ξ ) ≥ t dµ k,a ( ξ ) (18) is finite then for every f ∈ L pk,a ( R N ) we have (cid:18)Z R N (cid:16) |F k,a f ( ξ ) | ψ ( ξ ) b − p ′ (cid:17) b dµ k,a ( ξ ) (cid:19) b . M b − p ′ ϕ k f k L pk,a ( R N ) . (19)This naturally reduced to Hausdorff-Young inequality when b = p ′ and Paley inequality(10) when b = p. Proof.
From Theorem 2.1, the operator defined by Af ( ξ ) = F k,a f ( ξ ) , ξ ∈ R N is bounded from L pk,a ( R N ) to L p ( R N , ω dµ ′ ) , where dµ ′ ( ξ ) = dµ k,a ( ξ ) and ω ( ξ ) = ψ ( ξ ) − p . From Theorem 1.2, we deduce that A : L p ( R N ) → L p ′ ( R N , ω dµ ′ ) with dµ ′ ( ξ ) = dπ ( ξ )and ω ( ξ ) = 1 admits a bounded extension. By using the real interpolation (Theorem 2.2above) we will prove that A : L pk,a ( R N ) → L b ( R N , ωdµ ′ ) , p ≤ b ≤ p ′ , is bounded, wherethe space L p ( R N , ωdµ ′ ) is defined by the norm k σ k L p ( R N , ωdµ ′ ) := (cid:18)Z R N | σ ( ξ ) | p w ( ξ ) dµ ′ ( ξ ) (cid:19) p = (cid:18)Z R N | σ ( ξ ) | p w ( ξ ) dµ k,a ( ξ ) (cid:19) p and ω ( ξ ) is positive function over R N to be determined. To compute ω, we can useCorollary 2.2, by fixing θ ∈ (0 ,
1) such that b = − θp + θp ′ . In this case θ = p − bb ( p − , and ω = ω p (1 − θ ) p ω pθp = ψ ( ξ ) − bp ′ . (20)Thus we finish the proof. (cid:3) An operator A is a Fourier multiplier then there exists a measurable function h : R N → C , known as the symbol associated with A, such that F k,a ( Af )( ξ ) = h ( ξ ) F k,a f ( ξ ) , ξ ∈ R N , for all f belonging to a suitable function space on R N . In the next result, we show that ifthe symbol h of a Fourier multipliers A defined on C ∞ c ( R N ) satisfies certain H¨ormander’scondition then A can be extended as a bounded linear operator from L pk,a ( R N ) to L qk,a ( R N )for the range 1 < p ≤ ≤ q < ∞ . Theorem 2.4.
Let < p ≤ ≤ q < ∞ . Suppose that A is a Fourier multiplier withsymbol h. Then we have k A k L pk,a ( R N ) → L qk,a ( R N ) . sup s> s (cid:20)Z { ξ ∈ R N : | h ( ξ ) |≥ s } dµ k,a ( ξ ) (cid:21) p − q . Proof.
Let us first assume that p ≤ q ′ , where q + q ′ = 1 . Since q ′ ≤ , the Hausdorff-Younginequality gives that k Af k L qk,a ( R N ) ≤ kF k,a ( Af ) k L q ′ k,a ( R N ) = k h F k,a f k L q ′ k,a ( R N ) The case q ′ ≤ ( p ′ ) ′ = p can be reduced to the case p ≤ q ′ as follows. Using the dualityof L p -spaces we have k A k L pk,a ( R N ) → L q ( R N ) = k A ∗ k L q ′ k,a ( R N ) → L p ′ k,a ( R N ) . The symbol of adjointoperator A ∗ is equal to ˇ h, which equal to h and obviously we have | ˇ h | = | h | (see Theorem4.2 in [1]). Now, we are in a position to apply Theorem 2.3. Set p − q = r . Now, byapplying Theorem 2.3 with ψ = | h | r with b = q ′ we get k h F k,a f k L q ′ ( R + ,Adx ) . sup s> s Z ξ ∈ R N | h ( ξ ) | r>s dµ k,a ( ξ ) r k f k L pk,a ( R N ) for all f ∈ L pk,a ( R N ) , in view of p − q = q ′ − p ′ = r. Thus, for 1 < p ≤ ≤ q < ∞ , weobtain k Af k L qk,a ( R N ) . sup s> s Z ξ ∈ R N | h ( ξ ) | r>s dµ k,a ( ξ ) r k f k L pk,a ( R N ) . OUNDEDNESS OF ( k, a )-FOURIER MULTIPLIERS 11
Further, the proof follows from the following inequality: sup s> s Z ξ ∈ R N | h ( ξ ) | r>s dµ k,a ( ξ ) r = sup s> s Z ξ ∈ R N | h ( ξ ) | >s r dµ k,a ( ξ ) r = sup s> s r Z ξ ∈ R N | h ( ξ ) | >s dµ k,a ( ξ ) r = sup s> s Z ξ ∈ R N | h ( ξ ) | >s dµ k,a ( ξ ) r , proving Theorem 2.4. (cid:3) As an application of Theorem 2.4 we get the following result.
Corollary 2.5.
Let < γ < h k i + N + a − and let h be a measurable function on R N such that | h ( ξ ) | . k ξ k − γ , where k ξ k is the Euclidean norm of ξ ∈ R N . Then the ( k, a ) -Fourier multiplier T h withsymbol h is bounded from L pk,a ( R N ) to L qk,a ( R N ) provided that < p ≤ ≤ q < ∞ , p − q = γ h k i + N + a − . (21) Proof.
From Theorem 2.4 it follows that k A k L pk,a ( R N ) → L qk,a ( R N ) . sup s> s (cid:20)Z { ξ ∈ R N : | h ( ξ ) |≥ s } dµ k,a ( ξ ) (cid:21) p − q . sup s> s (cid:20)Z { ξ ∈ R N : s . k ξ k − γ } dµ k,a ( ξ ) (cid:21) p − q . Now, using the polar coordinates on R N and the fact that in polar coordinates it holdsthat dµ k,a ( x )(= v k,a ( x ) dx ) := r h k i + N + a − v k ( θ ) dr dσ ( θ ) (see [30]), we get k A k L pk,a ( R N ) → L qk,a ( R N ) . sup s> s (cid:20)Z { r ∈ R + : r . s − γ } r h k i + N + a − dr (cid:21) p − q . sup s> s h s − h k i + N + a − γ i ( p − q ) = sup s> < ∞ , by using the assumption (21). (cid:3) Applications to nonlinear PDEs
This section is devoted to the applications of our main result on L p - L q boundedness of( k, a )-Fourier multipliers to the well-posedness of abstract Cauchy problem on R N . Themethod we use here is similar to [14] in the case of the Fourier analysis associated to thebiorthogonal eigenfunction expansion of a model operator having discrete spectrum.3.1. Nonlinear Heat equation.
Let us consider the following Cauchy problem of non-linear evolution equation in the space L ∞ (0 , T, L k,a ( R N )) ,u t − | Bu ( t ) | p = 0 , u (0) = u , (22)where B is a linear operator on L k,a ( R N ) and 1 < p < ∞ . We say that the heat equation (22) admits a solution u if u ( t ) = u + Z t | Bu ( τ ) | p dτ (23)in the space L ∞ (0 , T, L pk,a ( R N )) for every T < ∞ . We say that u is a local solution of (22)if it satisfies the equation (23) in the space L ∞ (0 , T ∗ , L k,a ( R N )) for some T ∗ > . Theorem 3.1.
Let < p < ∞ . Suppose that B is Fourier multiplier such that its symbol h satisfies sup s> s (cid:20)Z { ξ ∈ R N : | h ( ξ ) |≥ s } dµ k,a ( ξ ) (cid:21) − p < ∞ . Then the Cauchy problem (22) has a local solution in the space L ∞ (0 , T ∗ , L k,a ( R N )) forsome T ∗ > . Proof.
By integrating equation (22) w.r.t. t one get u ( t ) = u + t Z | Bu ( τ ) | p dτ. By taking the L -norm on both sides, one obtains k u ( t ) k L k,a ( R N ) ≤ C k u k L k,a ( R N ) + (cid:13)(cid:13)(cid:13)(cid:13)Z t | Bu ( t ) | p dτ (cid:13)(cid:13)(cid:13)(cid:13) L k,a ( R N ) ! = C k u k L k,a ( R N ) + Z R N (cid:12)(cid:12)(cid:12)(cid:12)Z t | Bu ( t ) | p dτ (cid:12)(cid:12)(cid:12)(cid:12) dµ k,a ( x ) ! . OUNDEDNESS OF ( k, a )-FOURIER MULTIPLIERS 13
Using the inequality R t | Bu ( τ ) | p dτ ≤ ( R t dτ ) ( R t | Bu ( τ ) | p dτ ) = t ( R t | Bu ( τ ) | p dτ ) , we get k u ( t ) k L k,a ( R N ) ≤ C k u k L k,a ( R N ) + t Z R N Z t | Bu ( t ) | p dτ dµ k,a ( x ) ! ≤ C k u k L k,a ( R N ) + t Z t Z R N | Bu ( t ) | p dµ k,a ( x ) dτ ! ≤ C k u k L k,a ( R N ) + t Z t k Bu ( t ) k pL pk,a ( R N ) dτ ! . Next, using the condition on the symbol it can be seen, as an application of Theo-rem 2.4, that the operator B is a bounded operator from L k,a ( R N ) to L pk,a ( R N ), that is, k Bu ( t ) k L pk,a ( R N ) ≤ C k u ( t ) k L k,a ( R N ) and, therefore, the above inequality yields k u ( t ) k L k,a ( R N ) ≤ C k u k L k,a ( R N ) + t Z t k u ( t ) k pL k,a ( R N ) dτ ! , (24)for some constant C independent from u and t .Finally, by taking L ∞ -norm in time on both sides of the estimate (24), one obtains k u ( t ) k L ∞ (0 ,T ; L k,a ( R N )) ≤ C (cid:16) k u k L k,a ( R N ) + T k u k pL ∞ (0 ,T ; L k,a ( R N )) (cid:17) . (25)Let us introduce the following set S c := n u ∈ L ∞ (0 , T ; L k,a ( R N )) : k u k L ∞ (0 ,T ; L k,a ( R N )) ≤ c k u k L k,a ( R N ) o , (26)for some constant c ≥
1. Then, for u ∈ S c we have k u k L k,a ( R N ) + T k u k pL ∞ (0 ,T ; L k,a ( R N )) ≤ k u k L k,a ( R N ) + T c p k u k pL k,a ( R N ) . Finally, for u to be from the set S c it is enough to have, by invoking (25), that k u k L k,a ( R N ) + T c p k u k pL k,a ( R N ) ≤ c k u k L k,a ( R N ) . It can be obtained by requiring the following, T ≤ T ∗ := √ c − c p k u k L k,a ( R N ) . Thus, by applying the fixed point theorem, there exists a unique local solution u ∈ L ∞ (0 , T ∗ ; L k,a ( R N )) of the Cauchy problem (22). (cid:3) Nonlinear Wave Equation.
In this subsection, we will consider that the initialvalue problem for the nonlinear wave equation u tt ( t ) − b ( t ) | Bu ( t ) | p = 0 , (27)with the initial condition u (0) = u , u t (0) = u , where b is a positive bounded function depending only on time, B is a linear operator in L k,a ( R N ) and 1 < p < ∞ . We intend to study the well-posedness of the equation (27).We say that initial valued problem (27) admits a global solution u if it satisfies u ( t ) = u + tu + t Z ( t − τ ) b ( τ ) | Bu ( τ ) | p dτ (28)in the space L ∞ (0 , T ; L k,a ( R N )) for every T < ∞ .We say that (27) admits a local solution u if it satisfies the equation (28) in the space L ∞ k,a (0 , T ∗ ; L ( R N )) for some T ∗ > Theorem 3.2.
Let < p < ∞ . Suppose that B is a Fourier multiplier such that itssymbol h satisfies sup s> s (cid:20)Z { ξ ∈ R N : | h ( ξ ) |≥ s } dµ k,a ( ξ ) (cid:21) − p < ∞ . (i) If k b k L (0 ,T ) < ∞ for some T > then the Cauchy problem (27) has a local solutionin L ∞ (0 , T ; L k,a ( R N )) . (ii) Suppose that u is identically equal to zero. Let γ > / . Moreover, assumethat k b k L (0 ,T ) ≤ c T − γ for every T > , where c does not depend on T . Then,for every T > , the Cauchy problem (27) has a global solution in the space L ∞ (0 , T ; L k,a ( R N )) for sufficiently small u in L -norm.Proof. (i) By integrating the equation (27) two times in t one get u ( t ) = u + tu + t Z ( t − τ ) b ( τ ) | Bu ( τ ) | p dτ. By taking the L -norm on both sides, for t < T one obtains by simple calculation that OUNDEDNESS OF ( k, a )-FOURIER MULTIPLIERS 15 k u ( t ) k L k,a ( R N ) ≤ C k u k L k,a ( R N ) + t k u k L k,a ( R N ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) t Z ( t − τ ) b ( τ ) | Bu ( τ ) | p dτ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L k,a ( R N ) ≤ C k u k L k,a ( R N ) + t k u k L k,a ( R N ) + Z R N (cid:12)(cid:12)(cid:12) t Z ( t − τ ) b ( τ ) | Bu ( τ ) | p dτ (cid:12)(cid:12)(cid:12) dµ k,a ( x ) ≤ C k u k L k,a ( R N ) + t k u k L k,a ( R N ) + Z R N (cid:16) t t Z (cid:12)(cid:12)(cid:12) b ( τ ) | Bu ( τ ) | p (cid:12)(cid:12)(cid:12) dτ (cid:17) dµ k,a ( x ) ≤ C k u k L k,a ( R N ) + t k u k L k,a ( R N ) + Z R N t t Z (cid:12)(cid:12)(cid:12) b ( τ ) (cid:12)(cid:12)(cid:12) dτ t Z (cid:12)(cid:12)(cid:12) Bu ( τ ) (cid:12)(cid:12)(cid:12) p dτ dµ k,a ( x ) ≤ C k u k L k,a ( R N ) + t k u k L k,a ( R N ) + t k b k L (0 ,T ) Z R N t Z (cid:12)(cid:12)(cid:12) Bu ( τ ) (cid:12)(cid:12)(cid:12) p dτ dµ k,a ( x ) ≤ C k u k L k,a ( R N ) + t k u k L k,a ( R N ) + t k b k L (0 ,T ) t Z k Bu ( τ ) k pL pk,a ( R N ) dτ . Next, using the condition on the symbol it can be seen, as an application of Theorem2.4, that the operator B is a bounded operator from L k,a ( R N ) to L pk,a ( R N ), that is, k Bu ( t ) k L pk,a ( R N ) ≤ C k u ( t ) k L k,a ( R N ) and, therefore, the above inequality yields k u ( t ) k L k,a ( R N ) ≤ C ( k u k L k,a ( R N ) + t k u k L k,a ( R N ) + t k b k L (0 ,T ) t Z k u ( τ ) k pL pk,a ( R N ) dτ ) , (29)for some constant C not depending on u , u and t . Finally, by taking the L ∞ -norm intime on both sides of the estimate (29), one obtains k u k L ∞ (0 ,T ; L k,a ( R N )) ≤ C ( k u k L k,a ( R N ) + T k u k L k,a ( R N ) + T k b k L (0 ,T ) k u k pL ∞ (0 ,T ; L k,a ( R N )) ) . (30)Let us introduce the set S c := n u ∈ L ∞ (0 , T ; L k,a ( R N )) : k u k L ∞ (0 ,T ; L k,a ( R N )) ≤ c ( k u k L k,a ( R N ) + T k u k L k,a ( R N ) ) o (31)for some constant c ≥
1. Then, for u ∈ S c we have k u k L k,a ( R N ) + T k u k L k,a ( R N ) + T k b k L (0 ,T ) k u k pL ∞ (0 ,T ; L k,a ( R N )) ≤ k u k L k,a ( R N ) + T k u k L k,a ( R N ) + T k b k L (0 ,T ) c p (cid:16) k u k L k,a ( R N ) + T k u k L k,a ( R N ) (cid:17) p . (32)Observe that, to be u from the set S c it is enough to have, by invoking (30) and using(32), that k u k L k,a ( R N ) + T k u k L k,a ( R N ) + T k b k L (0 ,T ) c p (cid:16) k u k L k,a ( R N ) + T k u k L k,a ( R N ) (cid:17) p ≤ c ( k u k L k,a ( R N ) + T k u k L k,a ( R N ) ) . It can be obtained by requiring the following T ≤ T ∗ := min c − k b k L (0 ,T ) c p k u k p − L k,a ( R N ) , c − k b k L (0 ,T ) c p k u k p − L k,a ( R N ) . Thus, by applying the fixed point theorem, there exists a unique local solution u ∈ L ∞ (0 , T ∗ ; L k,a ( R N )) of the Cauchy problem (27).To prove Part (ii), we repeat the arguments of the proof of Part (i) to get (30). Now,by taking into account assumptions on u and b inequality (30) yields k u k L ∞ (0 ,T ; L k,a ( R N )) ≤ C (cid:16) k u k L k,a ( R N ) + T − γ k u k pL ∞ (0 ,T ; L k,a ( R N )) (cid:17) . (33)For a fixed constant c ≥
1, let us introduce the set S c := n u ∈ L ∞ (0 , T ; L k,a ( R N )) : k u k L ∞ (0 ,T ; L k,a ( R N )) ≤ cT γ k u k L k,a ( R N ) o , with γ > u ∈ S c we have k u k L k,a ( R N ) + T − γ k u k pL ∞ (0 ,T ; L k,a ( R N )) ≤ k u k L k,a ( R N ) + T − γ + γ p c p k u k pL k,a ( R N ) . To guarantee u ∈ S c , by invoking (33) we require that k u k L k,a ( R N ) + T − γ + γ p c p k u k pL k,a ( R N ) ≤ cT γ k u k L k,a ( R N ) . Now by choosing 0 < γ < γ − p such that ˜ γ := 3 − γ + γ p < , we obtain c p k u k p − L k,a ( R N ) ≤ cT − ˜ γ + γ . From the last estimate, we conclude that for any
T > k u k L k,a ( R N ) such that IVP (27) has a solution. It proves Part (ii) of Theorem 3.2. (cid:3) OUNDEDNESS OF ( k, a )-FOURIER MULTIPLIERS 17
Acknowledgment
VK and MR are supported by FWO Odysseus 1 grant G.0H94.18N: Analysis and PartialDifferential Equations. MR is also supported by the EPSRC Grant EP/R003025/1 andby the FWO grant G022821N.
References [1] R. Akylzhanov, E. Nursultanov and M. Ruzhansky. Hardy-Littlewood-Paley inequal-ities and Fourier multipliers on SU (2) . Studia Math.
234 (2016), no. 1, 1-29.[2] R. Akylzhanov, E. Nursultanov and M. Ruzhansky. Hardy-Littlewood, Hausdorff-Young-Paley inequalities, and L p − L q Fourier multipliers on compact homogeneousmanifolds.
J. Math. Anal. Appl.
479 (2019), no. 2, 1519–1548.[3] R. Akylzhanov and M. Ruzhansky. L p − L q multipliers on locally compact groups, J. Func. Anal. , 278(3) (2019), DOI: https://doi.org/10.1016/j.jfa.2019.108324[4] B. Amri and M. Gaidi. L p - L q estimates for the solution of the Dunkl wave equation, Manuscripta Math. , 159, 379-396 (2019).[5] J.-P. Anker, Fourier multipliers on Riemannian symmetric space of the noncompacttype,
Ann. of Math. (2) , 132(3) (1990) 597-628.[6] J.-P. Anker, An introduction to Dunkl theory and its analytic aspects. In Analytic,algebraic and geometric aspects of differential equations,
Trends Math. , pages 3–58.Birkh¨auser/Springer, Cham, 2017[7] S. Ben Sa¨ıd, T. Kobayashi and B. Ørsted, Laguerre semigroup and Dunkl operators,
Compos. Math. F k,a , C.R. Math. Acad. Sci. Paris , 347 (19-20) (2009), 1119-1124[9] S. Ben Sa¨ıd, A product formula and a convolution structure for a k -Hankel transformon R , J. Math. Anal. Appl. k, J . of Funct. Anal. 279(8) (2020), 108706.[11] J. Bergh and J. Lofstrom, Interpolation spaces , Grundlehren der mathematischenWissenschaften, (1976).[12] J. J. Betancor, A. J. Castro and J. Curbelo, Spectral multipliers for multidimensionalBessel operators,
J. Fourier Anal. Appl. L p on Ch´ebli-Trim`eche hypergroups, Proc. London Math. Soc. (3) L p – L q boundednessof pseudo-differential operators on smooth manifolds and its applications to nonlinearequations,(2020). https://arxiv.org/abs/2005.04936 [15] M. Chatzakou and V. Kumar, L p - L q boundedness of Fourier multipliers associatedwith the anharmonic Oscillator, (2020). https://arxiv.org/abs/2004.07801[16] J. L. Clere and E. M. Stein, L p -multipliers for non-compact symmetric spaces, Proc.Nat. Acad. Sci. U. S. A.
71 (1974) 3911-3913.[17] D. Constales, H. De Bie and P. Lian, Explicit formulas for the Dunkl dihedral kerneland the ( k, a )-generalized Fourier kernel,
J. Math. Anal. Appl. L p - L q estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces. I. Duke Math. J.
72 (1993), no.1, 109-150.[19] C. F. Dunkl, Differential-difference operators associated to reflection groups,
Trans.Am. Math. Soc.
Contemp.Math.
138 (1992) Providence, RI: Am. Math. Soc.)[21] H. De Bie, Clifford algebras, Fourier transforms, and quantum mechanics,
Mathe-matical Methods in the Applied Sciences
Invent Math
J. Fourier Anal. Appl.
19 (2013), no. 2, 417-437.[24] J. Dziuban´ski, and A. Hejna, Ho¨rmander’s multiplier theorem for the Dunkl trans-form.
J. Funct. Anal.
277 (2019), no. 7, 2133-2159.[25] D. V. Gorbachev, V. I. Ivanov and S. Yu. Tikhonov, Pitt’s inequalities and uncer-tainty principle for generalized Fourier transform,
Int. Math. Res. Not. IMRN , 23(2016) 7179-7200.[26] D. V. Gorbachev, V. I. Ivanov and S. Yu. Tikhonov, Positive L p -Bounded Dunkl-Type Generalized Translation Operator and Its Applications, Constructive approxi-mation , 49(3), (2019) 555-605.[27] J. Gosselin and K. Stempak, A weak-type estimate for Fourier-Bessel multipliers.
Proc. Amer. Math. Soc.
106 (1989), no. 3, 655-662.[28] L. H¨ormander, Estimates for translation invariant operators in L p spaces. Acta Math. ,104:93–140, 1960.[29] R. Howe, The oscillator semigroup, in the mathematical heritage of Hermann Weyl(Durham, NC 1987),
Proc. Symp. Pure Math. , 48, R.O. Wells, Ed. AMS Providence,1988.[30] T. R. Johansen, Weighted inequalities and uncertainty principles for the ( k, a )-generalized Fourier transform,
Internat. J. Math.
OUNDEDNESS OF ( k, a )-FOURIER MULTIPLIERS 19 [31] T. Kobayashi and G. Mano, Integral formula for the minimal representation of O ( p, , Acta. Appl. Math. O ( p, q ) , Proc. Japan. Acad. Ser. A Math. Sci. O ( p, q ), Mem. Amer. Math. Soc. , 213(1000)(2011), vi+132.[34] V. Kumar and M. Ruzhansky, Hardy-Littlewood inequality and L p – L q Fourier mul-tipliers on compact hypergroups, (2020). Arxiv: https://arxiv.org/abs/2005.08464.[35] H. Mejjaoli, Spectral theorems associated with the ( k, a )-generalized wavelet multi-pliers,
Journal of Pseudo-Differential Operators and Applications
Leuven, 2002 ), volume 1817 of
Lecture Notes in Math. , pages93–135. Springer, Berlin, 2003.[37] M. Ruzhansky and J. Wirth, L p Fourier multipliers on compact Lie groups.
Math.Z.
280 (2015), no. 3-4, 621-642.[38] F. Soltani, L p -Fourier multipliers for the Dunkl operator on the real line. J. Funct.Anal.
209 (2004), no. 1, 16-35.[39] K. Stempak, La th´eorie de Littlewood-Paley pour la transformation de Fourier-Ressel,
C. R. Acad. Sei. Paris
303 (1986), 15-18.[40] W. Teng, Hardy inequalities for fractional ( k, a )-generalized harmonic oscillator,(2020), https://arxiv.org/abs/2008.00804[41] B. Wr´obel, Multivariate spectral multipliers for the Dunkl transform and the Dunklharmonic oscillator,
Forum Math.
Vishvesh KumarDepartment of Mathematics: Analysis, Logic and Discrete MathematicsGhent University, Belgium
Email address : [email protected] Michael Ruzhansky:Department of Mathematics: Analysis, Logic and Discrete MathematicsGhent University, BelgiumandSchool of Mathematical SciencesQueen Mary University of London
United Kingdom