Fourier multipliers of Lipschitz spaces

Let G denote an infinite, compact, metrizable, 0-dimensional, Abelian group. The following are characterized: (i) the multipliers from one Lipschitz space Lip(α, p; G) to another Lipschitz space Lip(β, q; G) for 0 < α < β < ∞ and 1 ? p, q ? ∞; and (ii) the multipliers from Lip(α, p; G) to Lip(β, q; G) for 0 < β ? α < ∞ and 1 < q ? 2 ? p < ∞. Two special cases of (i), namely the case q = ∞ and the case p = 1, were obtained by the authors in an earlier publication (1981). A. Zygmund (J. Math. Mech.8 (1959), 889–895) and T. Mizuhara (Tôhoku Math. J.24 (1972), 263–268) have characterized the multipliers of certain Lipschitz spaces defined on the circle group.     

Fourier multipliers of classical modulation spaces

Based on the observation that translation invariant operators on modulation spaces are convolution operators we use techniques concerning pointwise multipliers for generalized Wiener amalgam spaces in order to give a complete characterization of the Fourier multipliers of modulation spaces. We deduce various applications, among them certain convolution relations between modulation spaces, as well as a short proof for a generalization of the main result of a recent paper by Bènyi et al., see [À. Bènyi, L. Grafakos, K. Gröchenig, K.A. Okoudjou, A class of Fourier multipliers for modulation spaces, Appl. Comput. Harmon. Anal. 19 (1) (2005) 131–139]. Finally, we show that any function with ([d/2]+1)-times bounded derivatives is a Fourier multiplier for all modulation spaces with p(1,∞) and q[1,∞].     

Unimodular Fourier multipliers for modulation spaces

We investigate the boundedness of unimodular Fourier multipliers on modulation spaces. Surprisingly, the multipliers with general symbol eiα|ξ|, where α∈[0,2], are bounded on all modulation spaces, but, in general, fail to be bounded on the usual Lp-spaces. As a consequence, the phase-space concentration of the solutions to the free Schrödinger and wave equations are preserved. As a byproduct, we also obtain boundedness results on modulation spaces for singular multipliers |ξ|^−δsin(α|ξ|) for 0?δ?α.     

A class of Fourier multipliers on starlike Lipschitz surfaces

In this paper, we consider a class of Fourier multipliers whose symbols are controlled by a polynomial on starlike Lipschitz surfaces and get the L² boundedness of these operators on Sobolev spaces and their endpoint estimates.     

Fourier transforms in generalized Lipschitz classes

We obtain sufficient conditions for the Fourier transform of a function f ∈ L ₁(?) to belong to generalized Lipschitz classes defined by the modulus of smoothness of order m. The sharpness of these conditions is established in the cases when f(t) ≥ 0 on ? or tf (t) ≥ 0 on ?.     

Fourier multipliers on the real Hardy spaces

We provide a variant of Hytönen’s embedding theorem, which allows us to extend and unify several sufficient conditions for a function to be a Fourier multiplier on the real Hardy spaces.     

Littlewood-paley operators on the generalized Lipschitz spaces

Littlewood-Paley operators defined on a new kind of generalized Lipschitz spaces ₀ ^,p are studied. It is proved that the image of a function under the action of these operators is either equal to infinity almost everywhere or is in ₀ ^,p , where –n<<1 and 1<p<.     

Fourier multipliers on power‐weighted Hardy spaces

Using Herz spaces, we obtain a sufficient condition for a bounded measurable function on ?ⁿ to be a Fourier multiplier on H^p_α (?ⁿ ) for 0 < p < 1 and –n < α ≤ 0. Our result is sharp in a certain sense and generalizes a recent result obtained by Baernstein and Sawyer. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fourier multipliers and spectral measures in Banach function spaces

It is classical that amongst all spaces L^p (G), 1 ≤ p ≤ ∞, for , or say, only L² (G) (that is, p = 2) has the property that every bounded Borel function on the dual group Γ determines a bounded Fourier multiplier operator in L² (G). Stone’s theorem asserts that there exists a regular, projection-valued measure (of operators on L² (G)), defined on the Borel sets of Γ, with Fourier-Stieltjes transform equal to the group of translation operators on L² (G); this fails for every p ≠ 2. We show that this special status of L² (G) amongst the spaces L^p (G), 1 ≤ p ≤ ∞, is actually more widespread; it continues to hold in a much larger class of Banach function spaces defined over G (relative to Haar measure).      

Fourier multipliers for Sobolev spaces on the Heisenberg group

In this paper, it is shown that the class of right Fourier multipliers for the Sobolev space W ^k,p (H ⁿ ) coincides with the class of right Fourier multipliers for L ^p (H ⁿ ) for k ∈ ?, 1 < p < ∞. Towards this end, it is shown that the operators R _j $ \bar R $ _j ?^?1 and $ \bar R $ _j R _j ?^?1 are bounded on L ^p (H ⁿ ), 1 < p < ∞, where $$ R_j = \frac{\partial } {{\partial z_j }} - \frac{i} {4}\bar z_j \frac{\partial } {{\partial t}}, \bar R_j = \frac{\partial } {{\partial \bar z_j }} + \frac{i} {4}z_j \frac{\partial } {{\partial t}} $$ and ? is the sublaplacian on H ⁿ . This proof is based on the Calderon-Zygmund theory on the Heisenberg group. It is also shown that when p = 1, the class of right multipliers for the Sobolev space W ^k,1(H ⁿ ) coincides with the dual space of the projective tensor product of two function spaces.     

Vector-valued Fourier multipliers on symmetric spaces of the noncompact type

LetX be a Riemannian symmetric space of the noncompact type. We prove the multiplier theorem for the Helgason-Fourier transform and the vector valued function spacesL _p (X, l _q ). As a consequence we get the inequalities of the Littlewood-Paley type forL _p (X) spaces.Research supported by K.B.N. Grant 210519101 (Poland).     

Fourier transforms and generalized Lipschitz classes in uniform metric

For functions f∈L¹(R)∩C(R) with Fourier transforms in L¹(R) we give necessary and sufficient conditions for f to belong to the generalized Lipschitz classes Hω,m and hω,m in terms of behavior of .     

1-Type Lipschitz selections in generalized 2-normed spaces

We shall introduce 1-type Lipschitz multifunctions from ℝ into generalized 2-normed spaces, and give some results about their 1-type Lipschitz selections.      

Coefficient multipliers of Lipschitz functions

Translated from Matematicheskie Zametki, Vol. 52, No. 5, pp. 68–77, November, 1992.