# $L^p$-$L^q$ multipliers on locally compact groups

@article{Akylzhanov2015LpLqMO, title={\$L^p\$-\$L^q\$ multipliers on locally compact groups}, author={Rauan Akylzhanov and Michael Ruzhansky}, journal={arXiv: Representation Theory}, year={2015} }

In this paper we discuss the $L^p$-$L^q$ boundedness of both spectral and Fourier multipliers on general locally compact separable unimodular groups $G$ for the range $1<p\leq q<\infty$. We prove a Lizorkin type multiplier theorem for $1<p\leq q<\infty$, and then refine it as a H\"ormander type multiplier theorem for $1<p\leq 2\leq q<\infty$. In the process, we establish versions of Paley and Hausdorff-Young-Paley inequalities on general locally compact separable unimodular groups. As a…

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## References

SHOWING 1-10 OF 76 REFERENCES

### Smooth Fourier multipliers in group algebras via Sobolev dimension

- Mathematics
- 2015

We investigate Fourier multipliers with smooth symbols defined over locally compact Hausdorff groups. Our main results in this paper establish new H\"ormander-Mikhlin criteria for spectral and…

### Sharpness in Young’s inequality for convolution

- Mathematics
- 1977

Let p and q be indices in the open interval (1, oo) such that pq < p + q; let r — pql(p + q — pq). It is shown here that there is a constant Cp,q < 1 such that, if G is a locally compact, unimodular…

### Lp FOURIER TRANSFORMS ON LOCALLY COMPACT UNIMODULAR GROUPS

- Mathematics
- 2010

We show among other things that (1) may be generalized as follows: Let G be a compact group with Haar measure 1 and let \cp\\ be any collection of inequivalent continuous irreducible unitary…

### Hypercontractivity in group von Neumann algebras

- Mathematics
- 2013

In this paper, we provide a combinatorial/numerical method to establish new hypercontractivity estimates in group von Neumann algebras. We will illustrate our method with free groups, triangular…

### Mean convergence of Fourier series on compact Lie groups

- Mathematics
- 1976

The main result is an L? mean convergence theorem for the partial sums of the Fourier series of a class function on a compact semisimple Lie group. A central element in the proof is a Lie group-Lie…

### AN EXTENSION OF PLANCHEREL'S FORMULA TO SEPARABLE UNIMODULAR GROUPS

- Mathematics
- 1950

We show that for any function f square-integrable on a separable unimodular locally compact group G (relative to Haar measure), the integral over G of the square of the absolute value of f equals the…

### A course in abstract harmonic analysis

- Mathematics
- 1995

Banach Algebras and Spectral Theory Banach Algebras: Basic Concepts Gelfand Theory Nonunital Banach Algebras The Spectral Theorem Spectral Theory of *-Representations Von Neumann Algebras Notes and…

### Pseudo-Differential Operators, Wigner Transform and Weyl Systems on Type I Locally Compact Groups

- MathematicsDocumenta Mathematica
- 2017

Let $G$ be a unimodular type I second countable locally compact group and $\hat G$ its unitary dual. We introduce and study a global pseudo-differential calculus for operator-valued symbols defined…

### Fourier multipliers on graded Lie groups

- Mathematics
- 2014

We study the $L^p$-boundedness of Fourier multipliers defined on graded nilpotent Lie groups via their group Fourier transform. We show that H\"ormander type conditions on the Fourier multipliers…

### Global quantization of pseudo-differential operators on compact Lie groups, SU(2), 3-sphere, and homogeneous spaces

- Mathematics
- 2013

Global quantization of pseudo-differential operators on general compact Lie groups G is introduced relying on the representation theory of the group rather than on expressions in local coordinates. A…