We give a quantitative characterization of the pairs of weights $(w,v)$ for which the dyadic version of the one-sided Hardy-Littlewood maximal operator satisfies a restricted weak $(p,p)$ type inequality for $1\le p<\infty$. More precisely, given any measurable set $E_0$, the estimate $$w\left(\{x\in {ℝ}^n:M^{+,d}\left({𝒳}_{E_0}\right)\left(x\right)>t\}\right)\le \frac{C{\left[(w,v)\right]}_{A_p^{+,d}\left(ℛ\right)}^p}{t^p}v\left(E_0\right)$$ holds if and only if the pair $(w,v)$ belongs to $A_p^{+,d}\left(ℛ\right)$, that is, $$\frac{\left|E\right|}{\left|Q\right|}\le {\left[(w,v)\right]}_{A_p^{+,d}\left(ℛ\right)}{\left(\frac{v\left(E\right)}{w\left(Q\right)}\right)}^{1/p}$$ for every dyadic cube $Q$ and every measurable set $E\subset Q^{+}$. The proof follows some ideas appearing in S. Ombrosi (2005). We also obtain a similar quantitative characterization for the non-dyadic...

In the first part we consider restriction theorems for hypersurfaces Γ in Rn, with the affine curvature KΓ1/(n+1) introduced as a mitigating factor. Sjölin, [19], showed that there is a universal restriction theorem for all convex curves in R2. We show that in dimensions greater than two there is no analogous universal restriction theorem for hypersurfaces with non-negative curvature.In the second part we discuss decay estimates for the Fourier transform of the density KΓ1/2 supported on the surface...

Let φ:ℝ² → ℝ be a homogeneous polynomial function of degree m ≥ 2, let Σ = (x,φ(x)): |x| ≤ 1 and let σ be the Borel measure on Σ defined by $\sigma \left(A\right)={\int }_B{\chi }_A(x,\phi \left(x\right))dx$ where B is the unit open ball in ℝ² and dx denotes the Lebesgue measure on ℝ². We show that the composition of the Fourier transform in ℝ³ followed by restriction to Σ defines a bounded operator from $L^p\left(ℝ³\right)$ to $L^q(\Sigma ,d\sigma )$ for certain p,q. For m ≥ 6 the results are sharp except for some border points.

Let $x\left(t\right)=(t,\frac{1}{2}{t}^2,\frac{1}{6}{t}^3)$ a certain curve in ${\mathbf{R}}^3.$ We investigate inequalities of the type $\{\int |\widehat{f}\left(x\right(t\left)\right){|}^{b}dt{\}}^{1/b}\le C\parallel f{\parallel }_a$ for $f\in 𝒮(\mathbf{R}$ 3). Our results improve improve an earlier restriction theorem of Prestini. Various generalizations are also discussed.

Some recent results on spline-Fourier and Ciesielski-Fourier series are summarized. The convergence of spline Fourier series and the basis properties of the spline systems are considered. Some classical topics, that are well known for trigonometric and Walsh-Fourier series, are investigated for Ciesielski-Fourier series, such as inequalities for the Fourier coefficients, convergence a.e. and in norm, Fejér and θ-summability, strong summability and multipliers. The connection between Fourier series...

In connection with the $A_{\varphi }$ classes of weights (see [K-T] and [B-K]), we study, in the context of Orlicz spaces, the corresponding reverse-Hölder classes $R{H}_{\varphi }$. We prove that when ϕ is ${\Delta }_2$ and has lower index greater than one, the class $R{H}_{\varphi }$ coincides with some reverse-Hölder class $R{H}_q,q>1$. For more general ϕ we still get $R{H}_{\varphi }\subset A_{\infty }={\bigcup }_{q>1}R{H}_q$ although the intersection of all these $R{H}_{\varphi }$ gives a proper subset of ${\bigcap }_{q>1}R{H}_q$.

We prove that ridgelet transform $R:𝒮\left({ℝ}^2\right)\to 𝒮\left(𝕐\right)$ and adjoint ridgelet transform $R^{*}:𝒮\left(𝕐\right)\to 𝒮\left({ℝ}^2\right)$ are continuous, where $𝕐={ℝ}^{+}\times ℝ\times [0,2\pi ]$. We also define the ridgelet transform $ℛ$ on the space ${𝒮}^{\text{'}}\left({ℝ}^2\right)$ of tempered distributions on ${ℝ}^2$, adjoint ridgelet transform ${ℛ}^{*}$ on ${𝒮}^{\text{'}}\left(𝕐\right)$ and establish that they are linear, continuous with respect to the weak${}^{*}$-topology, consistent with $R$, $R^{*}$ respectively, and they satisfy the identity $({ℛ}^{*}\circ ℛ)\left(u\right)=u$, $u\in {𝒮}^{\text{'}}\left({ℝ}^2\right)$.

Riemann’s memoir is devoted to the function π(x) defined as the number of prime numbers less or equal to the real and positive number x. This is really the fact, but the “main role” in it is played by the already mentioned zeta-function.

Elementary estimates for the Riesz kernel and for its derivative are given. Using these we show that the maximal operator of the Riesz means of a tempered distribution is bounded from $H_p\left(ℝ\right)$ to $L_p\left(ℝ\right)$ (1/(α+1) < p < ∞) and is of weak type (1,1), where $H_p\left(ℝ\right)$ is the classical Hardy space. As a consequence we deduce that the Riesz means of a function $⨍\in L_1\left(ℝ\right)$ converge a.e. to ⨍. Moreover, we prove that the Riesz means are uniformly bounded on $H_p\left(ℝ\right)$ whenever 1/(α+1) < p < ∞. Thus, in case $⨍\in H_p\left(ℝ\right)$, the Riesz means converge...

We show that the $L^p$ boundedness, p > 2, of the Riesz transform on a complete non-compact Riemannian manifold with upper and lower Gaussian heat kernel estimates is equivalent to a certain form of Sobolev inequality. We also characterize in such terms the heat kernel gradient upper estimate on manifolds with polynomial growth.

Given a set of positive measure on the circle and a set Λ of integers, one can ask whether $E\left(\Lambda \right):=e_{\lambda \in \Lambda }^{i\lambda t}$ is a Riesz sequence in L²(). We consider this question in connection with some arithmetic properties of the set Λ. Improving a result of Bownik and Speegle (2006), we construct a set such that E(Λ) is never a Riesz sequence if Λ contains an arithmetic progression of length N and step $\ell =O\left(N^{1-\epsilon }\right)$ with N arbitrarily large. On the other hand, we prove that every set admits a Riesz sequence E(Λ) such that Λ does contain...

We study the validity of the $L^p$ inequality for the Riesz transform when $p\&gt;2$ and of its reverse inequality when $1\&lt;p\&lt;2$ on complete riemannian manifolds under the doubling property and some Poincaré inequalities.