Related papers: $L^{p}$ estimates for bilinear and multi-parameter…

We investigate H\"ormander spectral multiplier theorems as they hold on $X = L^p(\Omega),\: 1 < p < \infty,$ for many self-adjoint elliptic differential operators $A$ including the standard Laplacian on $\R^d.$ A strengthened matricial…

We characterize two-weight inequalities for certain maximal truncations of the Hilbert transform in terms of testing conditions on simpler functions. For 1<p<2 and two positive Borel measures u, v on R, we assume that u is doubling, and we…

In this work we prove a new $L^p$ holomorphic extension result for functions defined on product Lipschitz surfaces with small Lipschitz constants in two complex variables. We define biparameter and partial Cauchy integral operators that…

We study a multilinear version of H\"ormander multiplier theorem, namely \begin{equation*} \Vert T_{\sigma}(f_1,\dots,f_n)\Vert_{L^p}\lesssim \sup_{k\in\mathbb{Z}}{\Vert…

Let $G$ be a locally compact unimodular group, and let $\phi$ be some function of $n$ variables on $G$. To such a $\phi$, one can associate a multilinear Fourier multiplier, which acts on some $n$-fold product of the non-commutative…

In this article, we address endpoint issues for the bilinear spherical maximal functions. We obtain borderline restricted weak type estimates for the well studied bilinear spherical maximal function…

Supplying the missing necessary conditions, we complete the characterisation of the $L^p\to L^q$ boundedness of commutators $[b,T]$ of pointwise multiplication and Calder\'on-Zygmund operators, for arbitrary pairs of $1<p,q<\infty$ and…

We prove the operator space Grothendieck inequality for bilinear forms on subspaces of noncommutative $L_p$-spaces with $2<p<\infty$. One of our results states that given a map $u: E\to F^*$, where $E, F\subset L_p(M)$ ($2<p<\infty$, $M$…

We characterize those bounded multilinear operators that factor through a Hilbert space in terms of its behavior in finite sequences. This extends a result, essentially due to S. Kwapie\'{n}, from the linear to the multilinear setting. We…

Given p between 1 and infinity, but not 2, we show that the T1 theorem for the Hilbert transform fails for L^{p}, despite holding for p equal to 2

We prove that the maximal operator associated with variable homogeneous planar curves $(t, u t^{\alpha})_{t\in \mathbb{R}}$, $\alpha\not=1$ positive, is bounded on $L^p(\mathbb{R}^2)$ for each $p>1$, under the assumption that…

The principle of optimizing inequalities, or their equivalent operator theoretic formulation, is well established in analysis. For an operator, this corresponds to extending its action to larger domains, hopefully to the largest possible…

Analogues of multi-paramter multiplier operators on R^d are defined on the torus T^d. It is shown that these operators satisfy the classical Coifman-Meyer theorem. In addition, L log L and L (log L)^n end-point estimates are proved.

For any Ritt operator T:L^{p}(\Omega) --> L^{p}(\Omega), for any positive real number \alpha, and for any x in L^{p}, we consider the square functions |x |_{T,\alpha} = \Bigl| \Bigl(\sum_{k=1}^{\infty} k^{2\alpha -1}\bigl…

We introduce and study a class of operator tuples in complex Hilbert spaces, which we call spherical tuples. In particular, we characterize spherical multi-shifts, and more generally, multiplication tuples on RKHS. We further use these…

In \cite{MR447956}, Muckenhoupt and Wheeden formulated a weighted weak $(p,p)$ inequality where the weight for the weak $L^p$ space is treated as a multiplier rather than a measure. They proved such inequalities for the Hardy-Littlewood…

Let $\mathcal{D}$ be the classical Dirichlet space, the Hilbert space of holomorphic functions on the disk. Given a holomorphic symbol function $b$ we define the associated Hankel type bilinear form, initially for polynomials f and g, by…

The $L^p$ boundedness theory of convolution operators is \linebreak based on an initial $L^2\to L^2$ estimate derived from the Fourier transform. The corresponding theory of multilinear operators lacks such a simple initial estimate in view…

For any natural number $k$, consider the $k$-linear Hilbert transform $$ H_k( f_1,\dots,f_k )(x) := \operatorname{p.v.} \int_{\bf R} f_1(x+t) \dots f_k(x+kt)\ \frac{dt}{t}$$ for test functions $f_1,\dots,f_k: {\bf R} \to {\bf C}$. It is…

This paper investigates the boundedness of bilinear pseudo-differential operators with symbols in the H\"{o}rmander class $BS_{\varrho,\delta}^m(\mathbb{R}^n)$ in the previously unexplored regime $0 \leq \varrho < \delta < 1$. We establish…