Theorems stating sufficient conditions for the inequivalence of the d-variate Haar wavelet system and another wavelet system in the spaces $L₁\left({ℝ}^d\right)$ and $BV\left({ℝ}^d\right)$ are proved. These results are used to show that the Strömberg wavelet system and the system of continuous Daubechies wavelets with minimal supports are not equivalent to the Haar system in these spaces. A theorem stating that some systems of smooth Daubechies wavelets are not equivalent to the Haar system in $L₁\left({ℝ}^d\right)$ is also shown.

It has been proved recently that the two-direction refinement equation of the form $f\left(x\right)={\sum }_{n\in }{c}_{n,1}f(kx-n)+{\sum }_{n\in \mathbb{Z}}{c}_{n,-1}f(-kx-n)$ can be used in wavelet theory for constructing two-direction wavelets, biorthogonal wavelets, wavelet packages, wavelet frames and others. The two-direction refinement equation generalizes the classical refinement equation $f\left(x\right)={\sum }_{n\in \mathbb{Z}}cₙf(kx-n)$, which has been used in many areas of mathematics with important applications. The following continuous extension of the classical refinement equation $f\left(x\right)={\int }_{ℝ}c\left(y\right)f(kx-y)dy$ has also various interesting...

The paper deals with local means and wavelet bases in weighted and unweighted function spaces of type $B_{pq}^s$ and $F_{pq}^s$ on ℝⁿ and on ⁿ.

We construct the Haar wavelets on a local field K of positive characteristic and show that the Haar wavelet system forms an unconditional basis for $L^p\left(K\right)$, 1 < p < ∞. We also prove that this system, normalized in $L^p\left(K\right)$, is a democratic basis of $L^p\left(K\right)$. This also proves that the Haar system is a greedy basis of $L^p\left(K\right)$ for 1 < p < ∞.

J. Bourgain, H. Brezis and P. Mironescu [in: J. L. Menaldi et al. (eds.), Optimal Control and Partial Differential Equations, IOS Press, Amsterdam, 2001, 439-455] proved the following asymptotic formula: if $\Omega \subset {ℝ}^d$ is a smooth bounded domain, 1 ≤ p < ∞ and $f\in W^{1,p}\left(\Omega \right)$, then $li{m}_{s\nearrow 1}(1-s){\int }_{\Omega }{\int }_{\Omega }{\left(\right|f\left(x\right)-f\left(y\right)|}^p{)/(\left|\right|x-y\left|\right|}^{d+sp})dxdy=K{\int }_{\Omega }{\left|\nabla f\left(x\right)\right|}^{p}dx$, where K is a constant depending only on p and d. The double integral on the left-hand side of the above formula is an equivalent seminorm in the Besov space $B_p^{s,p}\left(\Omega \right)$. The purpose of this paper is to obtain analogous asymptotic formulae...

Let $\Theta :={\theta }_I^e:e\in E,I\in D$ be a decomposition system for $L₂\left({ℝ}^d\right)$ indexed over D, the set of dyadic cubes in ${ℝ}^d$, and a finite set E, and let $\Theta ̃:=\Theta {̃}_I^e:e\in E,I\in D$ be the corresponding dual functionals. That is, for every $f\in L₂\left({ℝ}^d\right)$, $f={\sum }_{e\in E}{\sum }_{I\in D}⟨f,\Theta {̃}_I^e⟩{\theta }_I^e$. We study sufficient conditions on Θ,Θ̃ so that they constitute a decomposition system for Triebel-Lizorkin and Besov spaces. Moreover, these conditions allow us to characterize the membership of a distribution f in these spaces by the size of the coefficients $⟨f,\Theta {̃}_I^e⟩$, e ∈ E, I ∈ D. Typical examples of such decomposition...

Abstract We develop an almost orthogonal wavelet-type expansion in ℝ² which is adapted to polar coordinates. We start by defining a product Fourier-Hankel transform f̂ and proving a sampling formula for f such that f̂ is compactly supported. For general f, the sampling formula and a partition of unity lead to an identity of the form $f={\sum }_{\mu ,k,m}⟨f,{\phi }_{\mu km}⟩{\psi }_{\mu km}$, in which each function ${\phi }_{\mu km}$ and ${\psi }_{\mu km}$ is concentrated near a certain annular sector, has compactly supported product Fourier-Hankel transform, and is smooth...

We consider the subspace of L²(ℝ) spanned by the integer shifts of one function ψ, and formulate a condition on the family ${\psi (·-n)}_{n=-\infty }^{\infty }$, which is equivalent to the weight function ${\sum }_{n=-\infty }^{\infty }\left|\psi ̂(·+n)\right|²$ being > 0 a.e.

We study embeddings of spaces of Besov-Morrey type, $i{d}_{\Omega }{:}_{p₁,u₁,q₁}^{s₁}\left(\Omega \right){\hookrightarrow }_{p₂,u₂,q₂}^{s₂}\left(\Omega \right)$, where $\Omega \subset {ℝ}^d$ is a bounded domain, and obtain necessary and sufficient conditions for the continuity and compactness of $i{d}_{\Omega }$. This continues our earlier studies relating to the case of ${ℝ}^d$. Moreover, we also characterise embeddings into the scale of $L_p$ spaces or into the space of bounded continuous functions.

To each set of knots $t_i=i/2n$ for i = 0,...,2ν and $t_i=(i-\nu )/n$ for i = 2ν + 1,..., n + ν, with 1 ≤ ν ≤ n, there corresponds the space ${}_{\nu ,n}$ of all piecewise linear and continuous functions on I = [0,1] with knots $t_i$ and the orthogonal projection $P_{\nu ,n}$ of L²(I) onto ${}_{\nu ,n}$. The main result is $li{m}_{(n-\nu )\wedge \nu \to \infty }\left|\right|P_{\nu ,n}\left|\right|₁=su{p}_{\nu ,n:1\le \nu \le n}\left|\right|P_{\nu ,n}\left|\right|₁=2+(2-\surd 3)²$. This shows that the Lebesgue constant for the Franklin orthogonal system is 2 + (2-√3)².

It is known that Gabor expansions do not converge unconditionally in $L^p$ and that $L^p$ cannot be characterized in terms of the magnitudes of Gabor coefficients. By using a combination of Littlewood-Paley and Gabor theory, we show that $L^p$ can nevertheless be characterized in terms of Gabor expansions, and that the partial sums of Gabor expansions converge in $L^p$-norm.

We show that each general Haar system is permutatively equivalent in $L^p\left([0,1]\right)$, 1 < p < ∞, to a subsequence of the classical (i.e. dyadic) Haar system. As a consequence, each general Haar system is a greedy basis in $L^p\left([0,1]\right)$, 1 < p < ∞. In addition, we give an example of a general Haar system whose tensor products are greedy bases in each $L^p\left({[0,1]}^d\right)$, 1 < p < ∞, d ∈ ℕ. This is in contrast to [11], where it has been shown that the tensor products of the dyadic Haar system are not greedy bases...

The classical Hardy-Littlewood maximal operator is bounded not only on the classical Lebesgue spaces $L_p\left({ℝ}^d\right)$ (in the case $p>1$), but (in the case when $1/p(·)$ is log-Hölder continuous and $p_{-}=inf\{p\left(x\right):x\in {ℝ}^d\}>1$) on the variable Lebesgue spaces $L_{p(·)}\left({ℝ}^d\right)$, too. Furthermore, the classical Hardy-Littlewood maximal operator is of weak-type $(1,1)$. In the present note we generalize Besicovitch’s covering theorem for the so-called $\gamma$-rectangles. We introduce a general maximal operator $M_s^{\gamma ,\delta }$ and with the help of generalized $\Phi$-functions, the strong-...

The purpose of multifractal analysis of functions is to determine the Hausdorff dimensions of the sets of points where a function (or a distribution) f has a given pointwise regularity exponent H. This notion has many variants depending on the global hypotheses made on f; if f locally belongs to a Banach space E, then a family of pointwise regularity spaces $C_E^{\alpha }\left(x₀\right)$ are constructed, leading to a notion of pointwise regularity with respect to E; the case $E=L^{\infty }$ corresponds to the usual Hölder regularity,...

We study several techniques which are well known in the case of Besov and Triebel-Lizorkin spaces and extend them to spaces with dominating mixed smoothness. We use the ideas of Triebel to prove three important decomposition theorems. We deal with so-called atomic, subatomic and wavelet decompositions. All these theorems have much in common. Roughly speaking, they say that a function f belongs to some function space (say $S_{p,q}^{r̅}A$) if, and only if, it can be decomposed as $f\left(x\right)={\sum }_{\nu }{\sum }_m{\lambda }_{\nu m}{a}_{\nu m}\left(x\right)$, convergence in S’, with...

We prove three theorems on linear operators $T_{\tau ,p}:H_p\left(ℬ\right)\to H_p$ induced by rearrangement of a subsequence of a Haar system. We find a sufficient and necessary condition for $T_{\tau ,p}$ to be continuous for 0 < p < ∞.

Let $X={\sum }_{i=1}^k{a}_i{U}_i$, $Y={\sum }_{i=1}^k{b}_i{U}_i$, where the $U_i$ are independent random vectors, each uniformly distributed on the unit sphere in ℝⁿ, and $a_i,b_i$ are real constants. We prove that if $b{²}_i$ is majorized by $a{²}_i$ in the sense of Hardy-Littlewood-Pólya, and if Φ: ℝⁿ → ℝ is continuous and bisubharmonic, then EΦ(X) ≤ EΦ(Y). Consequences include most of the known sharp $L²-L^p$ Khinchin inequalities for sums of the form X. For radial Φ, bisubharmonicity is necessary as well as sufficient for the majorization inequality to always hold. Counterparts...

Let $M_S$ denote the strong maximal operator. Let $M_x$ and $M_y$ denote the one-dimensional Hardy-Littlewood maximal operators in the horizontal and vertical directions in ℝ². A function h supported on the unit square Q = [0,1]×[0,1] is exhibited such that ${\int }_Q{M}_y{M}_{x}h<\infty$ but ${\int }_Q{M}_x{M}_{y}h=\infty$. It is shown that if f is a function supported on Q such that ${\int }_Q{M}_y{M}_{x}f<\infty$ but ${\int }_Q{M}_x{M}_{y}f=\infty$, then there exists a set A of finite measure in ℝ² such that ${\int }_A{M}_{S}f=\infty$.

For an injective map τ acting on the dyadic subintervals of the unit interval [0,1) we define the rearrangement operator $T_s$, 0 < s < 2, to be the linear extension of the map $\left(h_I\right)/{\left(\right|I|}^{1/s})\mapsto \left(h_{\tau \left(I\right)}\right)\left(\right|\tau \left(I\right){|}^{1/s})$, where $h_I$ denotes the $L^{\infty }$-normalized Haar function supported on the dyadic interval I. We prove the following extrapolation result: If there exists at least one 0 < s₀ < 2 such that $T_{s₀}$ is bounded on $H^{s₀}$, then for all 0 < s < 2 the operator $T_s$ is bounded on $H^s$.

The classical criterion for compactness in Banach spaces of functions can be reformulated into a simple tightness condition in the time-frequency domain. This description preserves more explicitly the symmetry between time and frequency than the classical conditions. The result is first stated and proved for $L²\left({ℝ}^d\right)$, and then generalized to coorbit spaces. As special cases, we obtain new characterizations of compactness in Besov-Triebel-Lizorkin, modulation and Bargmann-Fock spaces. ...

We prove trace inequalities of type $\left|\right|u^{\text{'}}{\left|\right|}_{L^2}^2+{\sum }_{j\in \mathbb{Z}}{k}_j|u\left(a_j\right){|}^2{\ge \lambda \left|\right|u\left|\right|}_{L^2}^2$ where $u\in H^1\left(ℝ\right)$, under suitable hypotheses on the sequences ${a_j}_{j\in \mathbb{Z}}$ and ${k_j}_{j\in \mathbb{Z}}$, with the first sequence increasing and the second bounded.

We study the boundedness of the one-sided operator $g{⁺}_{\lambda ,\phi }$ between the weighted spaces $L^p\left(M¯w\right)$ and $L^p\left(w\right)$ for every weight w. If λ = 2/p whenever 1 < p < 2, and in the case p = 1 for λ > 2, we prove the weak type of $g{⁺}_{\lambda ,\phi }$. For every λ > 1 and p = 2, or λ > 2/p and 1 < p < 2, the boundedness of this operator is obtained. For p > 2 and λ > 1, we obtain the boundedness of $g{⁺}_{\lambda ,\phi }$ from $L^p\left({\left(M¯\right)}^{[p/2]+1}w\right)$ to $L^p\left(w\right)$, where ${\left(M¯\right)}^k$ denotes the operator M¯ iterated k times.

We prove sharp a priori estimates for the distribution function of the dyadic maximal function ℳ ϕ, when ϕ belongs to the Lorentz space $L^{p,q}$, 1 < p < ∞, 1 ≤ q < ∞. The approach rests on a precise evaluation of the Bellman function corresponding to the problem. As an application, we establish refined weak-type estimates for the dyadic maximal operator: for p,q as above and r ∈ [1,p], we determine the best constant $C_{p,q,r}$ such that for any $\varphi \in L^{p,q}$, ${\left|\right|ℳ\varphi \left|\right|}_{r,\infty }\le C_{p,q,r}{\left|\right|\varphi \left|\right|}_{p,q}$.

Suppose Δ̃ is the Laplace-Beltrami operator on the sphere $S^{d-1},{\Delta }_{\rho }^{k}f\left(x\right)={\Delta }_{\rho }{\Delta }_{\rho }^{k-1}f\left(x\right)$ and ${\Delta }_{\rho }f\left(x\right)=f\left(\rho x\right)-f\left(x\right)$ where ρ ∈ SO(d). Then ${\omega }^m{(f,t)}_{L_p\left(S^{d-1}\right)}\equiv sup\parallel {\Delta }_{\rho }^{m}f{\parallel }_{L_p\left(S^{d-1}\right)}:\rho \in SO\left(d\right),ma{x}_{x\in S^{d-1}}\rho x·x\ge cost$ and $K̃ₘ{(f,t^m)}_p\equiv inf\parallel f-g{\parallel }_{L_p\left(S^{d-1}\right)}+t^m\parallel {(-\Delta ̃)}^{m/2}g{\parallel }_{L_p\left(S^{d-1}\right)}:g\in \left({(-\Delta ̃)}^{m/2}\right)$ are equivalent for 1 < p < ∞. We note that for even m the relation was recently investigated by the second author. The equivalence yields an extension of the results on sharp Jackson inequalities on the sphere. A new strong converse inequality for $L_p\left(S^{d-1}\right)$ given in this paper plays a significant role in the proof.

We prove some results in the context of isoperimetric inequalities with quantitative terms. In the $2$-dimensional case, our main contribution is a method for determining the optimal coefficients $c_1,...,c_m$ in the inequality $\delta P\left(E\right)\ge {\sum }_{k=1}^m{c}_k\alpha {\left(E\right)}^k+o\left(\alpha {\left(E\right)}^m\right)$, valid for each Borel set $E$ with positive and finite area, with $\delta P\left(E\right)$ and $\alpha \left(E\right)$ being, respectively, the $\mathrm{𝑖𝑠𝑜𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑟𝑖𝑐𝑑𝑒𝑓𝑖𝑐𝑖𝑡}$ and the $\mathrm{𝐹𝑟𝑎𝑒𝑛𝑘𝑒𝑙𝑎𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑦}$ of $E$. In $n$ dimensions, besides proving existence and regularity properties of minimizers for a wide class of $\mathrm{𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑎𝑡𝑖𝑣𝑒𝑖𝑠𝑜𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑟𝑖𝑐𝑞𝑢𝑜𝑡𝑖𝑒𝑛𝑡𝑠}$ including the lower semicontinuous extension of $\frac{\delta P\left(E\right)}{\alpha {\left(E\right)}^2}$, we...

For n ∈ ℕ, L > 0, and p ≥ 1 let ${\kappa }_p(n,L)$ be the largest possible value of k for which there is a polynomial P ≢ 0 of the form $P\left(x\right)={\sum }_{j=0}^n{a}_j{x}^j$, $|a_0\left|\ge L\right({\sum }_{j=1}^n|a_j{{|}^p)}^{1/p}$, $a_j\in ℂ$, such that ${(x-1)}^k$ divides P(x). For n ∈ ℕ, L > 0, and q ≥ 1 let ${\mu }_q(n,L)$ be the smallest value of k for which there is a polynomial Q of degree k with complex coefficients such that $\left|Q\left(0\right)\right|>1/L({\sum }_{j=1}^n{\left|Q\left(j\right)\right|}^q{)}^{1/q}$. We find the size of ${\kappa }_p(n,L)$ and ${\mu }_q(n,L)$ for all n ∈ ℕ, L > 0, and 1 ≤ p,q ≤ ∞. The result about ${\mu }_{\infty }(n,L)$ is due to Coppersmith and Rivlin, but our proof is completely different and much shorter even...

An RD-space $𝒳$ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds. The authors prove that for a space of homogeneous type $𝒳$ having “dimension” $n$, there exists a $p_0\in (n/(n+1),1)$ such that for certain classes of distributions, the $L^p\left(𝒳\right)$ quasi-norms of their radial maximal functions and grand maximal functions are equivalent when $p\in (p_0,\infty ]$. This result yields a radial maximal function characterization for Hardy spaces on $𝒳$. ...

The main aim of this paper is to prove that the maximal operator ${\sigma }^{}$ is bounded from the Hardy space $H_{1/2}$ to weak-$L_{1/2}$ and is not bounded from $H_{1/2}$ to $L_{1/2}$.

We prove several optimal Moser–Trudinger and logarithmic Hardy–Littlewood–Sobolev inequalities for systems in two dimensions. These include inequalities on the sphere $S^2$, on a bounded domain $\Omega \subset {ℝ}^2$ and on all of ${ℝ}^2$. In some cases we also address the question of existence of minimizers.

In this paper we study the regularity properties of the one-dimensional one-sided Hardy-Littlewood maximal operators ${ℳ}^{+}$ and ${ℳ}^{-}$. More precisely, we prove that ${ℳ}^{+}$ and ${ℳ}^{-}$ map $W^{1,p}\left(ℝ\right)\to W^{1,p}\left(ℝ\right)$ with $1<p<\infty$, boundedly and continuously. In addition, we show that the discrete versions $M^{+}$ and $M^{-}$ map $\mathrm{BV}\left(\mathbb{Z}\right)\to \mathrm{BV}\left(\mathbb{Z}\right)$ boundedly and map $l^1\left(\mathbb{Z}\right)\to \mathrm{BV}\left(\mathbb{Z}\right)$ continuously. Specially, we obtain the sharp variation inequalities of $M^{+}$ and $M^{-}$, that is, $$\mathrm{Var}\left(M^{+}\left(f\right)\right)\le \mathrm{Var}\left(f\right)\text{and}\mathrm{Var}\left(M^{-}\left(f\right)\right)\le \mathrm{Var}\left(f\right)$$ if $f\in \mathrm{BV}\left(\mathbb{Z}\right)$, where $\mathrm{Var}\left(f\right)$ is the total variation of $f$ on $\mathbb{Z}$ and $\mathrm{BV}\left(\mathbb{Z}\right)$ is the set of all functions $f:\mathbb{Z}\to ℝ$ satisfying $\mathrm{Var}\left(f\right)<\infty$.

Let $f^j={\sum }_k{a}_{k}f(2^{j+1}x-2k)$, where the sum is taken over the lattice of all points k in ${ℝ}^n$ having integer-valued components, j∈ℕ and $a_k\in ℂ$. Let $A_{pq}^s$ be either $B_{pq}^s$ or $F_{pq}^s$ (s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞) on ${ℝ}^n.$ The aim of the paper is to clarify under what conditions $\parallel f^j|A_{pq}^s\parallel$ is equivalent to $2^{j(s-n/p)}({\sum }_k|a_k{|}^p{)}^{1/p}\parallel f|A_{pq}^s\parallel$.

A space $X$ is said to have the Rothberger property (or simply $X$ is Rothberger) if for every sequence $\langle {𝒰}_n:n\in \omega \rangle$ of open covers of $X$, there exists $U_n\in {𝒰}_n$ for each $n\in \omega$ such that $X={\bigcup }_{n\in \omega }{U}_n$. For any $n\in \omega$, necessary and sufficient conditions are obtained for $C_p{(\Psi \left(𝒜\right),2)}^n$ to have the Rothberger property when $𝒜$ is a Mrówka mad family and, assuming CH (the Continuum Hypothesis), we prove the existence of a maximal almost disjoint family $𝒜$ for which the space $C_p{(\Psi \left(𝒜\right),2)}^n$ is Rothberger for all $n\in \omega$.