We introduce the higher order Lipschitz classes Λr(α) and λr(α) of periodic functions by means of the rth order difference operator, where r = 1, 2, ..., and 0 < α ≦ r. We study the smoothness property of a function f with absolutely convergent Fourier series and give best possible sufficient conditions in terms of its Fourier coefficients
in order that f belongs to one of the above classes.
:= [−π, π). We give sufficient conditions in terms of the coefficients in order that >f belong to one of the multiplicative Lipschitz classes Lip (α1,..., αN) and lip (α1,..., αN) for some α1,..., αN > 0. These multiplicative Lipschitz classes of functions are defined in terms of the multiple difference operator of first
order in each variable. The conditions given by us are not only sufficient, but also necessary for a special subclass of coefficients.
Our auxiliary results on the equivalence between the order of magnitude of the rectangular partial sums and that of the rectangular
remaining sums of related N-multiple numerical series may be useful in other investigations, too.
We consider complex-valued functions f∊L1(ℝ+), where ℝ+:=[0,∞), and prove sufficient conditions under which the sine Fourier transform and the cosine Fourier transform belong to one of the Lipschitz classes Lip (α) and lip (α) for some 0<α≦1, or to one of the Zygmund classes Zyg (α) and zyg (α) for some 0<α≦2. These sufficient conditions are best possible in the sense that they are also necessary if f(x)≧0 almost everywhere.
Summary A multivariate Hausdorff operator H = H(µ, c, A) is defined in terms of a s-finite Borel measure µ on Rn, a Borel measurable function c on Rn, and an n × n matrix A whose entries are Borel measurable functions on rn and such that A is nonsingular µ-a.e. The operator H*:= H (µ, c | det A-1|, A-1) is the adjoint to H in a well-defined sense. Our goal is to prove sufficient conditions for the boundedness of these operators on the real Hardy space H1(Rn) and BMO (Rn). Our main tool is proving commuting relations among H, H*, and the Riesz transforms Rj. We also prove commuting relations among H, H*, and the Fourier transform.
<a name="abs1"/>Abstract??We give sufficient conditions for the convergence of the double Fourier integral of a complex-valued functionf?L1(?2) with bounded support at a given point (x0,y0) ? ?2. It turns out that this convergence essentially depends on the convergence of the single Fourier integrals of the marginal functionsf(x,y0),x? ?, andf(x0,y),y? ?, at the pointsx:=x0andy:=y0, respectively. Our theorem applies to functions in the multiplicative Zygmund classes of functions in two variables.
We prove sufficient conditions for the convergence of the integrals conjugate to the double Fourier integral of a complex-valued
function f ∈ L1 (ℝ2) with bounded support at a given point (x0, g0) ∈ ℝ2. It turns out that this convergence essentially depends on the convergence of the integral conjugate to the single Fourier
integral of the marginal functions f(x, y0), x ∈ ℝ, and f(x0, y), y ∈ ℝ, at x:= x0 and y:= y0, respectively. Our theorems apply to functions in the multiplicative Lipschitz and Zygmund classes introduced in this paper.