The main purpose of this paper is to prove Hörmander’s $L^p$–$L^q$ boundedness of Fourier multipliers on commutative hypergroups. We carry out this objective by establishing the Paley inequality and Hausdorff–Young–Paley inequality for commutative hypergroups. We show the $L^p$–$L^q$ boundedness of the spectral multipliers for the generalised radial Laplacian by examining our results on Chébli–Trimèche hypergroups. As a consequence, we obtain embedding theorems and time asymptotics for the $L^p$–$L^q$ norms of the heat kernel for generalised radial Laplacian.

We prove convergence in norm and pointwise almost everywhere on $L^p$, $p\in (1,\infty )$, for certain multi-parameter polynomial ergodic averages by establishing the corresponding multi-parameter maximal and oscillation inequalities. Our result, in particular, gives an affirmative answer to a multi-parameter variant of the Bellow–Furstenberg problem. This paper is also the first systematic treatment of multi-parameter oscillation semi-norms which allows an efficient handling of multi-parameter pointwise convergence problems with arithmetic features. The methods of proof of our main result develop estimates for multi-parameter exponential sums, as well as introduce new ideas from the so-called multi-parameter circle method in the context of the geometry of backwards Newton diagrams that are dictated by the shape of the polynomials defining our ergodic averages.

In this paper, we mainly investigate the well-posedness of the four-order degenerate differential equation ($P_4$): $(Mu)''''(t) + \alpha (Lu)'''(t) + (Lu)''(t)$ $=\beta Au(t) + \gamma Bu'(t) + Gu'_t + Fu_t + f(t),\,( t\in [0,\,2\pi ])$ in periodic Lebesgue–Bochner spaces $L^p(\mathbb {T}; X)$ and periodic Besov spaces $B_{p,q}^s\;(\mathbb {T}; X)$, where $A$, $B$, $L$ and $M$ are closed linear operators on a Banach space $X$ such that $D(A)\cap D(B)\subset D(M)\cap D(L)$ and $\alpha,\,\beta,\,\gamma \in \mathbb {C}$, $G$ and $F$ are bounded linear operators from $L^p([-2\pi,\,0];X)$ (respectively $B_{p,q}^s([-2\pi,\,0];X)$) into $X$, $u_t(\cdot ) = u(t+\cdot )$ and $u'_t(\cdot ) = u'(t+\cdot )$ are defined on $[-2\pi,\,0]$ for $t\in [0,\, 2\pi ]$. We completely characterize the well-posedness of ($P_4$) in the above two function spaces by using known operator-valued Fourier multiplier theorems.

We investigate norms of spectral projectors on thin spherical shells for the Laplacian on tori. This is closely related to the boundedness of resolvents of the Laplacian and the boundedness of $L^{p}$ norms of eigenfunctions of the Laplacian. We formulate a conjecture and partially prove it.

Let $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}\in \mathbb{R}$ and $s\in \mathbb{N}$ be given. Let $\unicode[STIX]{x1D6FF}_{x}$ denote the Dirac measure at $x\in \mathbb{R}$, and let $\ast$ denote convolution. If $\unicode[STIX]{x1D707}$ is a measure, $\unicode[STIX]{x1D707}^{\star }$ is the measure that assigns to each Borel set $A$ the value $\overline{\unicode[STIX]{x1D707}(-A)}$. If $u\in \mathbb{R}$, we put $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},u}=e^{iu(\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D6FD})/2}\unicode[STIX]{x1D6FF}_{0}-e^{iu(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD})/2}\unicode[STIX]{x1D6FF}_{u}$. Then we call a function $g\in L^{2}(\mathbb{R})$ a generalized$(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})$-difference of order$2s$ if for some $u\in \mathbb{R}$ and $h\in L^{2}(\mathbb{R})$ we have $g=[\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},u}+\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},u}^{\star }]^{s}\ast h$. We denote by ${\mathcal{D}}_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},s}(\mathbb{R})$ the vector space of all functions $f$ in $L^{2}(\mathbb{R})$ such that $f$ is a finite sum of generalized $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})$-differences of order $2s$. It is shown that every function in ${\mathcal{D}}_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},s}(\mathbb{R})$ is a sum of $4s+1$ generalized $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})$-differences of order $2s$. Letting $\widehat{f}$ denote the Fourier transform of a function $f\in L^{2}(\mathbb{R})$, it is shown that $f\in {\mathcal{D}}_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},s}(\mathbb{R})$ if and only if $\widehat{f}$  “vanishes” near $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ at a rate comparable with $(x-\unicode[STIX]{x1D6FC})^{2s}(x-\unicode[STIX]{x1D6FD})^{2s}$. In fact, ${\mathcal{D}}_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},s}(\mathbb{R})$ is a Hilbert space where the inner product of functions $f$ and $g$ is $\int _{-\infty }^{\infty }(1+(x-\unicode[STIX]{x1D6FC})^{-2s}(x-\unicode[STIX]{x1D6FD})^{-2s})\widehat{f}(x)\overline{\widehat{g}(x)}\,dx$. Letting $D$ denote differentiation, and letting $I$ denote the identity operator, the operator $(D^{2}-i(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD})D-\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}I)^{s}$ is bounded with multiplier $(-1)^{s}(x-\unicode[STIX]{x1D6FC})^{s}(x-\unicode[STIX]{x1D6FD})^{s}$, and the Sobolev subspace of $L^{2}(\mathbb{R})$ of order $2s$ can be given a norm equivalent to the usual one so that $(D^{2}-i(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD})D-\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}I)^{s}$ becomes an isometry onto the Hilbert space ${\mathcal{D}}_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},s}(\mathbb{R})$. So a space ${\mathcal{D}}_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},s}(\mathbb{R})$ may be regarded as a type of Sobolev space having a negative index.

We show that the discrete Hilbert transform and the discrete Kak–Hilbert transform are infinitesimal generators of one-parameter groups of operators in ${{\ell }^{2}}$.

We prove weak-type $\left( 1,\,1 \right)$ estimates for compositions of maximal operators with singular integrals. Our main object of interest is the operator $\Delta *\Psi $ where $\Delta *$ is Bourgain’s maximal multiplier operator and $\Psi $ is the sum of several modulated singular integrals; here our method yields a significantly improved bound for the ${{L}^{q}}$ operator norm when $1\,<\,q\,<\,2$. We also consider associated variation-norm estimates.

It is considered whether $ L\,{=}\,\limsup_{n\to\infty} n \snormo{T^{n+1}-T^n}\,\lt\,\infty$ implies that the operator $T$ is power-bounded. It is shown that this is so if $L\lt1/e$, but it does not necessarily hold if $L\,{=}\,1/e$. As part of the methods, a result of Esterle is improved, showing that if $\sigma(T)\,{=}\,\{1\}$ and $T\,{\ne}\,I$, then $\liminf_{n\to\infty} n \snormo{T^{n+1}-T^n} \ge 1/e$. The constant $1/e$ is sharp. Finally, a way to create many generalizations of Esterle's result is described, and also many conditions are given on an operator which imply that its norm is equal to its spectral radius.

Let $X$ be a complex Banach space and let ${{B}_{p}}\left( X \right)$ denote the vector-valued Bergman space on the unit disc for $1\,\le \,p\,<\,\infty $. A sequence ${{\left( {{T}_{n}} \right)}_{n}}$ of bounded operators between two Banach spaces $X$ and $Y$ defines a multiplier between ${{B}_{p}}\left( X \right)$ and ${{B}_{q}}\left( Y \right)$ (resp. ${{B}_{p}}\left( X \right)$ and ${{l}_{q}}\left( Y \right)$) if for any function $f\left( z \right)\,=\,\sum _{n=0}^{\infty }\,{{x}_{n}}{{z}^{n}}$ in ${{B}_{p}}\left( X \right)$ we have that $g\left( z \right)\,=\,\sum _{n=0}^{\infty }\,{{T}_{n}}\left( {{x}_{n}} \right){{z}^{n}}$ belongs to ${{B}_{q}}\left( Y \right)$ (resp. ${{\left( {{T}_{n}}\left( {{x}_{n}} \right) \right)}_{n}}\,\in \,{{\ell }_{q}}\left( Y \right)$). Several results on these multipliers are obtained, some of them depending upon the Fourier or Rademacher type of the spaces $X$ and $Y$. New properties defined by the vector-valued version of certain inequalities for Taylor coefficients of functions in ${{B}_{p}}\left( X \right)$ are introduced.

We study when the spaces of Lorentz multipliers from ${{L}^{p,t\,}}\to \,{{L}^{p,s}}$ are distinct. Our main interest is the case when $s\,<\,t$, the Lorentz-improving multipliers. We prove, for example, that the space of multipliers which map ${{L}^{p,t\,}}\to \,{{L}^{p,s}}$ is different from those mapping ${{L}^{r,v}}\,\to \,{{L}^{r,u}}$ if either $r\,=\,p$ or ${p}'$ and $1/s\,-\,1/t\,\ne \,1/u\,-\,1/v$, or $r\,\ne \,p$ or ${p}'$. These results are obtained by making careful estimates of the Lorentz multiplier norms of certain linear combinations of Fejer or Dirichlet kernels. For the case when the first indices are different the linear combination we analyze is in the spirit of a Rudin-Shapiro polynomial.

In the paper we find, for certain values of the parameters, the spaces of multipliers (H(p, q, α), H(s, t, β) and (H(p, q, α), ls), where H(p, q, α) denotes the space of analytic functions on the unit disc such that . As corollaries we recover some new results about multipliers on Bergman spaces and Hardy spaces.

Let S2jf be the 2j-th partial sum of the Vilenkin-Fourier series of f ∊ L1, and set S2-1f = 0. For , we show that the ratio

is contained between two bounds (independent of f) . From this we obtain the Marcinkiewicz multiplier theorem for Vilenkin-Fourier series.

Let X be a closed subspace of LP(μ), where μ is an arbitrary measure and 1 < p < ∞. By extending the scope of spectral integration, we show that every invertible power-bounded linear mapping of X into X has a functional calculus implemented by the algebra of complex-valued functions on the unit circle satisfying the hypotheses of the Strong Marcinkiewicz Multiplier Theorem. This result expands the framework of the Strong Marcinkiewicz Multiplier Theorem to the setting of abstract measure spaces.

We present a new elementary proof of the fact that a norm one multiplier ϕ on LP(T) satisfying ϕ(0) = ϕ(k) = 1 is k-periodic, and extend this result, when possible, to multipliers on translation invariant subspaces of LP. A consequence of our work is that all such multipliers on HP(T) are the restriction of a norm one multiplier on LP(T).

We continue to develop the theory of previous papers concerning transforms corresponding to Mellin multipliers which involve products and/or quotients of Γ-functions. We show that, by working with certain subspaces of Lp,μ consisting of smooth functions, we can simplify considerably the restrictions on the parameters which were necessary in the Lp,μ setting. As a result, operators in our class become homeomorphisms on these subspaces under conditions of great generality.

Necessary multiplier conditions for Laguerre expansions are derived and discussed within the framework of weighted Lebesgue spaces.