The spaces of bilinear multipliers of weighted Lorentz type modulation spaces

Fix a nonzero window ${g\in\mathcal{S}(\mathbb{R}^{n})}$, a weight function w on ${\mathbb{R}^{2n}}$ and ${1\leq p,q\leq\infty}$. The weighted Lorentz type modulation space ${M(p,q,w)(\mathbb{R}^{n})}$ consists of all tempered distributions ${f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})}$ such that the short time Fourier transform ${V_{g}f}$ is in the weighted Lorentz space ${L(p,q,w\,d\mu)(\mathbb{R}^{2n})}$. The norm on ${M(p,q,w)(\mathbb{R}^{n})}$ is ${\|f\/\|_{M(p,q,w)}=\|V_{g}f\/\|_{pq,w}}$. This space was firstly defined and some of its properties were investigated for the unweighted case by Gürkanlı in [9] and generalized to the weighted case by Sandıkçı and Gürkanlı in [16]. Let ${1<p_{1},p_{2}<\infty}$, ${1\leq q_{1},q_{2}<\infty}$, ${1\leq p_{3},q_{3}\leq\infty}$, ${\omega_{1},\omega_{2}}$ be polynomial weights and ${\omega_{3}}$ be a weight function on ${\mathbb{R}^{2n}}$. In the present paper, we define the bilinear multiplier operator from ${M(p_{1},q_{1},\omega_{1})(\mathbb{R}^{n})\times M(p_{2},q_{2},\omega_{2})(% \mathbb{R}^{n})}$ to ${M(p_{3},q_{3},\omega_{3})(\mathbb{R}^{n})}$ in the following way. Assume that ${m(\xi,\eta)}$ is a bounded function on ${\mathbb{R}^{2n}}$, and define$B_{m}(f,g)(x)=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\hat{f}(\xi)\hat{g}(% \eta)m(\xi,\eta)e^{2\pi i\langle\xi+\eta,x\rangle}\,d\xi\,d\eta\quad\text{for % all ${f,g\in\mathcal{S}(\mathbb{R}^{n})}$. }$The function m is said to be a bilinear multiplier on ${\mathbb{R}^{n}}$ of type ${(p_{1},q_{1},\omega_{1};p_{2},q_{2},\omega_{2};p_{3},q_{3},\omega_{3})}$ if ${B_{m}}$ is the bounded bilinear operator from ${M(p_{1},q_{1},\omega_{1})(\mathbb{R}^{n})\times M(p_{2},q_{2},\omega_{2})(% \mathbb{R}^{n})}$ to ${M(p_{3},q_{3},\omega_{3})(\mathbb{R}^{n})}$. We denote by ${\mathrm{BM}(p_{1},q_{1},\omega_{1};p_{2},q_{2},\omega_{2})(\mathbb{R}^{n})}$ the space of all bilinear multipliers of type ${(p_{1},q_{1},\omega_{1};p_{2},q_{2},\omega_{2};p_{3},q_{3},\omega_{3})}$, and define ${\|m\|_{(p_{1},q_{1},\omega_{1};p_{2},q_{2},\omega_{2};p_{3},q_{3},\omega_{3})% }=\|B_{m}\|}$. We discuss the necessary and sufficient conditions for ${B_{m}}$ to be bounded. We investigate the properties of this space and we give some examples.

SHARP CONSTANTS BETWEEN EQUIVALENT NORMS IN WEIGHTED LORENTZ SPACES

For an increasing weight w in Bp (or equivalently in Ap), we find the best constants for the inequalities relating the standard norm in the weighted Lorentz space Λp(w) and the dual norm.

On the Boundedness of Classical Operators on Weighted Lorentz Spaces

Conditions on weights 𝑢(·), υ(·) are given so that a classical operator T sends the weighted Lorentz space Lrs (υd𝑥) into Lpq (υd𝑥). Here T is either a fractional maximal operator Mα or a fractional integral operator Iα or a Calderón–Zygmund operator. A characterization of this boundedness is obtained for Mα and Iα when the weights have some usual properties and max(r, s) ≤ min(p, q).

Boundedness of Some Integral Operators

We apply the expression for the norm of a function in the weighted Lorentz space, with respect to the distribution function, to obtain as a simple consequence some weighted inequalities for integral operators.