$L^p$-$L^q$ boundedness of $(k, a)$-Fourier multipliers with applications to Nonlinear equations

The $(k,a)$-generalised Fourier transform is the unitary operator defined using the $a$-deformed Dunkl harmonic oscillator.The main aim of this paper is to prove $L^p$-$L^q$ boundedness of $(k, a)$-generalised Fourier multipliers. To show the boundedness we first establish Paley inequality and Hausdorff-Young-Paley inequality for $(k, a)$-generalised Fourier transform. We also demonstrate applications of obtained results to study the well-posedness of nonlinear partial differential equations.


Introduction and Basics on (k, a)-generalised Fourier transform
In his seminal paper [28], Hörmander initiated the study of boundedness of the translation invariant operators on R N . The translation invariant operators on R N can be characterised using the classical Euclidean Fourier transform on R N and therefore they are also known as Fourier multipliers. The boundedness of Fourier multipliers is useful to solve problems in the area of mathematical analysis, in particular, in PDEs. Hörmander [28] established the L p -boundedness and L p -L q boundedness of Fourier multipliers on R N . After that, L p -boundedness of Fourier 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 transform and very recently in [24] in the multidimensional setting. Recently, the researchers have turned their attention to establish the boundedness of L p -L q multipliers for the range 1 < p ≤ 2 ≤ q < ∞, see [1,3,4,14,15,18,34].
Precisely, the second author and his collaborators started investigating the Hörmander L p -L q Fourier multipliers theorem and its different consequences for locally compact groups and on homogeneous manifolds. Such analysis includes the Hardy-Littlewood inequality, spectral multipliers theorems and applications to PDEs [1,3,2,34]. In [15], similar results have been proved for the eigenfunction expansions of anharmonic oscillators and extended to the more general setting of bi-orthogonal expansions in [14]. Ben Saïd et al. [7,8] introduced (k, a)-generalised Fourier transform. It generalises many important integral transforms including Fourier transform and Dunkl transform on the Euclidean spaces R N [9,8]. Recently, there is a growing interest to develop the analysis related to the (k, a)-generalised Fourier transform. Notably, the uncertainty principles and Pitt inequalities [25,30], maximal function and translation operator [10], wavelets multipliers [35] and Hardy inequality [40] were explored by many researchers. In this paper, we establish L p -L q boundedness of (k, a)-Fourier multipliers using the (k, a)-generalised Fourier transform.
The proof of the main result hinges upon the Paley inequality and Hausdorff-Young-Palay inequality for (k, a)-generalised Fourier transform obtained by using the Hausdorff-Young inequality established in [30,25].
To describe our main result let us recall the classical Hörmander Fourier multipliers theorem settled in [28]: has a bounded extension from L p (R N ) to L q (R N ) provided that the symbol m satisfies the condition Here x, ξ denotes the standard Euclidean inner product of two vectors x and ξ in R N and x will denote the Euclidean norm on R N . The Euclidean Fourier transform F on R N can be described using the spectral information of the harmonic oscillator ∆ where ∆ R N is the Laplacian on R N . In fact, Howe [29] found the following description of the Euclidean Fourier transform F : This description has been proved to be useful to define generalisations of the Fourier transform such as Clifford algebra-valued Fourier transform and fractional Fourier transform. These constructions have been explained in an excellent overview article [21]. On the other hand, Dunkl [19,20] presented a generalisation of the Euclidean Fourier transform 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 the root 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. We set To describe the Dunkl Laplacian, let us define the first order Dunkl operator for ξ ∈ R N and for a fixed multiplicity function k by where ∂ ξ is the direction derivation in the direction of ξ and R + denotes the positive root subsystem. Let us fix an orthonormal basis {ξ 1 , ξ 2 , . . . , ξ N } for the inner product space (R N , ·, · ) and write T ξ j (k) as T j (k) for j ∈ {1, 2, . . . , N}. Then the Dunkl Laplacian is defined by ∆ k = N j=1 T j (k) 2 . The Dunkl Laplacian has explicit form and also plays a very important 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 just the Euclidean Fourier transform F and the Euclidean Laplacian ∆ R N , respectively. Using the Dunkl Laplacian one can define the Dunkl harmonic oscillator (or Dunkl-Hermite operator) as ∆ k − x 2 . Ben Saïd et al. [7] considered the a-deformed Dunkl harmonic oscillator given by ∆ k,a := x 2−a ∆ k − x a , a > 0.
By making use of this a-deformed Dunkl harmonic oscillator ∆ k,a , they introduced a two parameters unitary operator, (k, a)-generalised Fourier transform, F k,a on L 2 k,a (R N ), by F k,a := exp iπ 2 The (k, a)-generalised Fourier transform F k,a includes some prominent transforms on the Euclidean space R N : • For a = 2 and k > 0, F k,a is the Dunkl transform [22].
• For a = 2 and k ≡ 0, F k,a is the Euclidean Fourier transform [29].
• For a = 1 and k ≡ 0, F k,a is the Hankel transform appearing as the unitary inversion operator of the Schrödinger model of the minimal representation of the group O(N + 1, 2) (see [31,32,33]).
For a > 0 and a + 2 k + N − 2 > 0, the (k, a)-generalised Fourier transform F k,a is a bijective linear operator such that By the Schwartz kernel theorem there exists a distribution kernel B k,a (ξ, x) such that  (ii) a = 1 and 2 k + N − 2 ≥ 0 (iii) a = 2; (iii) k = 0 and a = 2 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 a finite constant that depends only on N, k, and a.
The following result is the Hausdorff -Young inequality for (k, a)-generalised Fourier transform.
Theorem 1.2. [30, Proposition 2.9] Assume that N, k, and a satisfy the assumption of It was conjectured by Gorbachev et al. [25] that if a + 2 k + N − 3 ≥ 0 then the kernel Hausdorff-Young inequality (5) becomes 1.
From this point onward, we always assume that N, k and a either satisfy assumptions of mentioning it explicitly. In fact, our results will hold if we assume that N ≥ 1, k ≥ 0 and a > 0 are such that a + 2 k + N − 2 > 0 and the distribution kernel B k,a is uniformly With having all the basics of (k, a)-generalised Fourier transform we are now in a position 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 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 establish this result is the following Hausdorff-Young Paley inequality for (k, a)-generalised Fourier Next, we will present applications of our main results in the context of well-posedness of nonlinear abstract Cauchy problems in the space L ∞ (0, T, L 2 k,a (R N )). First, we consider the heat equation where B is a linear operator on L 2 k,a (R N ) and 1 < p < ∞. We study local well-posedness of the heat equation (7) above. Secondly, we consider the initial value problem for the nonlinear wave equation with the initial condition u(0) = u 0 , u t (0) = u 1 , where b is a positive bounded function depending only on time, B is a linear operator in L 2 k,a (R N ) and 1 < p < ∞. We explore the 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 present the proof of Paley inequality and Hausdorff-Young-Paley inequality. Then, we give the proof of our main result concerning the L p -L q boundedness of (k, a)-Fourier multipliers and its consequences. In the last section, the applications of the results obtained in previous section will be discussed.

Main results
Throughout the paper, we shall use the notation A B to indicate A ≤ cB for a suitable constant c > 0. In this section, we will present our main results. In the proofs we follow 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 Then for f ∈ L p k,a (R N ), 1 < p ≤ 2, we have Proof. Let us consider a measure ν k,a on R N given by We define the corresponding L p (R N , ν k,a )-space, 1 ≤ p < ∞, as the space of all complexvalued function f defined by R N such that We will show that the sublinear operator T : is well-defined and bounded from L p k,a (R N ) to L p (R N , ν k,a ) for any 1 < p ≤ 2. In other words, we claim the following estimate: which will give us the required inequality (10) with M ψ := sup t>0 t ξ∈R N ψ(ξ)≥t dµ k,a (ξ). We will show that T is weak-type (2, 2) and weak-type (1, 1). More precisely, with the distribution function, where ν k,a is given by formula (11), we show that Then the estimate (12) follows from the Marcinkiewicz interpolation Theorem. Now, to show (13), using the Plancherel identity we get Thus, T is type (2, 2) with norm M 2 ≤ 1. Further, we show that T is of weak type (1, 1) with norm M 1 = M ψ ; more precisely, we show that The left hand side of (15) is an integral ψ(ξ) 2 dµ k,a (ξ) taken over all those ξ ∈ R N for for any y > 0 and, therefore, Now by setting w := Now we claim that Indeed, first we notice that for some c > 0. By interchanging the order of integration we get Further, by making substitution τ = t 2 , it gives This establishes our claim (17) and eventually proves (15). Therefore, we have proved (13) and (14). Then by using the Marcinkiewicz interpolation theorem with p 1 = 1 and p 2 = 2 This completes the proof of the theorem.
Next we record the following interpolation theorem from [11] for further use.

Now, we use the previous theorem to establish the Hausdorff-Young-Paley inequality using the interpolation between Hausdorff-Young inequality and Paley inequality for
(k, a)-generalised Fourier transform.
Theorem 2.3. Let 1 < p ≤ 2, and let 1 < p ≤ b ≤ p ′ < ∞, where p ′ = p p−1 . If ψ is a positive function on R N such that is finite then, for every f ∈ L p k,a (R N ), we have This naturally reduced to the Hausdorff-Young inequality (5) when b = p ′ and to the Paley inequality (10) when b = p.
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 for all f belonging to a suitable function space on R N . In the next result, we show that if the symbol h of a Fourier multipliers A defined on C ∞ c (R N ) satisfies certain Hörmander's condition then A can be extended as a bounded linear operator from L p k,a (R N ) to L q k,a (R N ) for the range 1 < p ≤ 2 ≤ q < ∞. Theorem 2.4. Let 1 < p ≤ 2 ≤ q < ∞. Suppose that A is a Fourier multiplier with symbol h. Then we have Proof. Let us first assume that p ≤ q ′ , where 1 q + 1 q ′ = 1. Since q ′ ≤ 2, the Hausdorff-Young inequality gives that The case q ′ ≤ (p ′ ) ′ = p can be reduced to the case p ≤ q ′ as follows. Using the duality . The symbol of adjoint operator A * is equal toȟ, which equal to h and obviously we have |ȟ| = |h| (see Theorem 4.2 in [1]). Now, we are in a position to apply Theorem 2.3. Set 1 Further, the proof follows from the following inequality: proving Theorem 2.4.
Remark 1. For a = 2 and k ≡ 0, we recover the classical theorem of Hörmander [28] on L p -L q boundedness of Fourier multipliers on R N as in this case F k,a and µ k,a become the Euclidean Fourier transform and the Lebesgue measure on R N , respectively.
As an application of Theorem 2.4 we get the following result.
Corollary 2.5. Let 0 < γ < 2 k + N + a − 2 and let h be a measurable function on R N such that where ξ is the Euclidean norm of ξ ∈ R N . Then the (k, a)-Fourier multiplier T h with symbol h is bounded from L p k,a (R N ) to L q k,a (R N ) provided that Proof. It follows from Theorem 2.4 that Now, using the polar coordinates on R N and the fact that in polar coordinates it holds that dµ k,a (x)(= v k,a (x) dx) := r 2 k +N +a−3 v k (θ) dr dσ(θ) (see [30]), we get by using the assumption (21).

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 . The method we use here is the same as in [14] for the case of the Fourier analysis associated to the biorthogonal eigenfunction expansion of a model operator on smooth manifolds having discrete spectrum.

Nonlinear Heat equation.
Let us consider the following Cauchy problem of nonlinear evolution equation in the space L ∞ (0, T, L 2 k,a (R N )), where B is a linear operator on L 2 k,a (R N ) and 1 < p < ∞. We say that the heat equation (22) admits a solution u if in the space L ∞ (0, T, L p k,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 2 k,a (R N )) for some T * > 0.
Then the Cauchy problem (22) has a local solution in the space L ∞ (0, T * , L 2 k,a (R N )) for some T * > 0.
Proof. By integrating equation (22) w.r.t. t one get By taking the L 2 -norm on both sides, one obtains Using the inequality Next, using the condition on the symbol h it can be seen, as an application of Theorem 2.4, that the operator B is a bounded operator from L 2 k,a (R N ) to L 2p k,a (R N ), that is, k,a (R N ) ≤ C 1 u(t) L 2 k,a (R N ) and, therefore, the above inequality yields for some constant C independent from u 0 and t.
Finally, by taking L ∞ -norm in time on both sides of the estimate (24), one obtains Let us introduce the following set for some constant c ≥ 1. Then, for u ∈ S c we have Finally, for u to be from the set S c it is enough to have, by invoking (25), that It can be obtained by requiring the following, Thus, by applying the fixed point theorem, there exists a unique local solution u ∈ L ∞ (0, T * ; L 2 k,a (R N )) of the Cauchy problem (22).

Nonlinear Wave Equation.
In this subsection, we will consider that the initial value problem for the nonlinear wave equation with the initial condition where b is a positive bounded function depending only on time, B is a linear operator in L 2 k,a (R N ) and 1 < p < ∞. We intend to study the well-posedness of the wave equation (27).
We say that the initial value problem (27) admits a global solution u if it satisfies in the space L ∞ (0, T ; L 2 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 Proof. (i) By integrating the equation (27) two times in t one get By taking the L 2 -norm on both sides, for t < T one obtains by simple calculation that Next, using the condition on the symbol it can be seen, as an application of Theorem 2.4, that the operator B is a bounded operator from L 2 k,a (R N ) to L 2p k,a (R N ), that is, k,a (R N ) ≤ C 1 u(t) L 2 k,a (R N ) and, therefore, the above inequality yields k,a (R N ) dτ ), (29) for some constant C not depending on u 0 , u 1 and t. Finally, by taking the L ∞ -norm in time on both sides of the estimate (29), one obtains L ∞ (0,T ;L 2 k,a (R N )) ).
Let us introduce the set S c := u ∈ L ∞ (0, T ; L 2 k,a (R N )) : u 2 for some constant c ≥ 1. Then, for u ∈ S c we have Observe that, to be u from the set S c it is enough to have, by invoking (30) and using (32), that Thus, by applying the fixed point theorem, there exists a unique local solution u ∈ L ∞ (0, T * ; L 2 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 1 and b inequality (30) yields For a fixed constant c ≥ 1, let us introduce the set S c := u ∈ L ∞ (0, T ; L 2 k,a (R N )) : u 2 L ∞ (0,T ;L 2 k,a (R N )) ≤ cT γ 0 u 0 2 L 2 k,a (R N ) , with γ 0 > 0 is to be defined later. Now, note that for u ∈ S c we have To guarantee u ∈ S c , by invoking (33) we require that From the last estimate, we conclude that for any T > 0 there exists sufficiently small u 0 L 2 k,a (R N ) such that IVP (27) has a solution. It proves Part (ii) of Theorem 3.2.