An Area Theorem for Joint Harmonic Functions on the Product of Homogeneous Trees

For harmonic functions v on the disc, it has been known for a long time that non-tangential boundedness a.e.is equivalent to finiteness a.e. of the integral of the area function of v (Lusin area theorem). This result also hold for functions that are non-tangentially bounded only in a measurable subset of the boundary, and has been extended to rank-one hyperbolic spaces, and also to infinite trees (homogeneous or not). No equivalent of the Lusin area theorem is known on higher rank symmetric spaces, with the exception of the degenerate higher rank case given by the cartesian product of rank-one hyperbolic spaces. Indeed, for products of two discs, an area theorem for jointly harmonic functions was proved by M.P. and P. Malliavin, who introduced a new area function; non-tangential boundedness a.e. is a sufficient condition, but not necessary, for the finiteness of this area integral. Their result was later extended to general products of rank-one hyperbolic spaces by Korányi and Putz. Here we prove an area theorem for jointly harmonic functions on the product of a finite number of infinite homogeneous trees; for the sake of simplicity, we give the proofs for the product of two trees. This could be the first step to an area theorem for Bruhat–Tits affine buildings, thereby shedding light on the higher rank continuous set-up.

Fourier restriction in low fractal dimensions

Let $S \subset \mathbb {R}^{n}$ be a smooth compact hypersurface with a strictly positive second fundamental form, $E$ be the Fourier extension operator on $S$ , and $X$ be a Lebesgue measurable subset of $\mathbb {R}^{n}$ . If $X$ contains a ball of each radius, then the problem of determining the range of exponents $(p,q)$ for which the estimate $\| Ef \|_{L^{q}(X)} \lesssim \| f \|_{L^{p}(S)}$ holds is equivalent to the restriction conjecture. In this paper, we study the estimate under the following assumption on the set $X$ : there is a number $0 < \alpha \leq n$ such that $|X \cap B_R| \lesssim R^{\alpha }$ for all balls $B_R$ in $\mathbb {R}^{n}$ of radius $R \geq 1$ . On the left-hand side of this estimate, we are integrating the function $|Ef(x)|^{q}$ against the measure $\chi _X \,{\textrm {d}}x$ . Our approach consists of replacing the characteristic function $\chi _X$ of $X$ by an appropriate weight function $H$ , and studying the resulting estimate in three different regimes: small values of $\alpha$ , intermediate values of $\alpha$ , and large values of $\alpha$ . In the first regime, we establish the estimate by using already available methods. In the second regime, we prove a weighted Hölder-type inequality that holds for general non-negative Lebesgue measurable functions on $\mathbb {R}^{n}$ and combine it with the result from the first regime. In the third regime, we borrow a recent fractal Fourier restriction theorem of Du and Zhang and combine it with the result from the second regime. In the opposite direction, the results of this paper improve on the Du–Zhang theorem in the range $0 < \alpha < n/2$ .

Approximations in $$L^1$$ with convergent Fourier series

For a separable finite diffuse measure space $${\mathcal {M}}$$ M and an orthonormal basis $$\{\varphi _n\}$$ { φ n } of $$L^2({\mathcal {M}})$$ L 2 ( M ) consisting of bounded functions $$\varphi _n\in L^\infty ({\mathcal {M}})$$ φ n ∈ L ∞ ( M ) , we find a measurable subset $$E\subset {\mathcal {M}}$$ E ⊂ M of arbitrarily small complement $$|{\mathcal {M}}{\setminus } E|<\epsilon $$ | M \ E | < ϵ , such that every measurable function $$f\in L^1({\mathcal {M}})$$ f ∈ L 1 ( M ) has an approximant $$g\in L^1({\mathcal {M}})$$ g ∈ L 1 ( M ) with $$g=f$$ g = f on E and the Fourier series of g converges to g, and a few further properties. The subset E is universal in the sense that it does not depend on the function f to be approximated. Further in the paper this result is adapted to the case of $${\mathcal {M}}=G/H$$ M = G / H being a homogeneous space of an infinite compact second countable Hausdorff group. As a useful illustration the case of n-spheres with spherical harmonics is discussed. The construction of the subset E and approximant g is sketched briefly at the end of the paper.

Integrals of subharmonic functions and their differences with weight over small sets on a ray

Let $E$ be a measurable subset in a segment $[0,r]$ in the positive part of the real axis in the complex plane, and $U=u-v$ be the difference of subharmonic functions $u\not\equiv -\infty$ and $v\not\equiv -\infty$ on the complex plane. An integral of the maximum on circles centered at zero of $U^+:=\sup\{0,U\} $ or $|u|$ over $E$ with a function-multiplier $g\in L^p(E) $ in the integrand is estimated, respectively, in terms of the characteristic function $T_U$ of $U$ or the maximum of $u$ on circles centered at zero, and also in terms of the linear Lebesgue measure of $E$ and the $ L^p$-norm of $g$. Our main theorem develops the proof of one of the classical theorems of Rolf Nevanlinna in the case $E=[0,R]$, given in the classical monograph by Anatoly A. Goldberg and Iossif V. Ostrovsky, and also generalizes analogs of the Edrei\,--\,Fuchs Lemma on small arcs for small intervals from the works of A.\,F.~Grishin, M.\,L.~Sodin, T.\,I.~Malyutina. Our estimates are uniform in the sense that the constants in these estimates do not depend on $U$ or $u$, provided that $U$ has an integral normalization near zero or $u(0)\geq 0$, respectively.

The thermodynamic formalism and exponents of singularity of invariant measure of circle maps with a single break

Let $T \in C^{2+ \varepsilon} (S^{1} \setminus \{x_{b} \})$, $\varepsilon> 0 $, be a circle homeomorphism with one break point $x_{b}$, at which $ T'(x) $ has a discontinuity of the first kind and both one-sided derivatives at the point $x_{b} $ are strictly positive. Assume that the rotation number $\rho_{T}$ is irrational and its decomposition into a continued fraction beginning from a certain place coincides with the golden mean, i.e., $\rho_{T}=[m_{1}, m_{2}, \ldots, m_{l}, \, m_{l + 1}, \ldots] $, $ m_{s} = 1$, $s> l> 0$. Since the rotation number is irrational, the map $ T $ is strictly ergodic, that is, possesses a unique probability invariant measure $\mu_{T}$. A.A. Dzhalilov and K.M. Khanin proved that the probability invariant measure $ \mu_{G} $ of any circle homeomorphism $ G \in C^{2+ \varepsilon} (S^{1} \setminus \{x_{b} \})$, $\varepsilon> 0$, with one break point $ x_{b} $ and the irrational rotation number $ \rho_{G} $ is singular with respect to the Lebesgue measure $ \lambda $ on the circle, i.e., there is a measurable subset of $ A \subset S^{1} $ such that $ \mu_ {G} (A) = 1 $ and $ \lambda (A) = 0$. We will construct a thermodynamic formalism for homeomorphisms $ T_{b} \in C^{2+ \varepsilon} (S^{1} \setminus \{x_{b} \})$, $\varepsilon> 0 $, with one break at the point $ x_{b} $ and rotation number equal to the golden mean, i.e., $ \rho_{T}:= \frac {\sqrt{5} -1}{2} $. Using the constructed thermodynamic formalism, we study the exponents of singularity of the invariant measure $ \mu_{T} $ of homeomorphism $ T $.

McShane Integrability Using Variational Measure

If f : [a, b] → R is McShane integrable on [a, b], then f is McShane integrable on every Lebesgue measurable subset of [a, b]. However, integrability of a real-valued function on [a, b] does not imply McShane integrability on any E ⊆ [a, b]. In this paper, we give a characterization for the McShane integrability of f : [a, b] → R over E ⊆ [a, b] using concept of variational measure.

Density topologies on the plane between ordinary and strong. II

Let C0 denote a set of all non-decreasing continuous functions f : (0, 1] → (0, 1] such that limx→0+f(x) = 0 and f(x) ≤ x for every x ∊ (0, 1], and let A be a measurable subset of the plane. The notions of a density point of A with respect to f and the mapping defined on the family of all measurable subsets of the plane were introduced in Wagner-Bojakowska, E. Wilcziński, W.: Density topologies on the plane between ordinary and strong, Tatra Mt. Math. Publ. 44 (2009), 139 151. This mapping is a lower density, so it allowed us to introduce the topology Tf , analogously to the density topology. In this note, properties of the topology Tf and functions approximately continuous with respect to f are considered. We prove that (ℝ2, Tf) is a completely regular topological space and we study conditions under which topologies generated by two functions f and g are equal.

QUASI-RANDOM PROFINITE GROUPS

Inspired by Gowers' seminal paper (W. T. Gowers, Comb. Probab. Comput.17(3) (2008), 363–387, we will investigate quasi-randomness for profinite groups. We will obtain bounds for the minimal degree of non-trivial representations of SLk(ℤ/(pnℤ)) and Sp2k(ℤ/(pnℤ)). Our method also delivers a lower bound for the minimal degree of a faithful representation of these groups. Using the suitable machinery from functional analysis, we establish exponential lower and upper bounds for the supremal measure of a product-free measurable subset of the profinite groups SLk(ℤp) and Sp2k(ℤp). We also obtain analogous bounds for a special subgroup of the automorphism group of a regular tree.

RATIONAL TIME-FREQUENCY GABOR FRAMES ASSOCIATED WITH PERIODIC SUBSETS OF THE REAL LINE

For a, b > 0 and g ∈ L2(ℝ), write 𝒢(g, a, b) for the Gabor system: [Formula: see text] Let S be an aℤ-periodic measurable subset of ℝ with positive measure. It is well-known that the projection 𝒢(gχS, a, b) of a frame 𝒢(g, a, b) in L2(ℝ) onto L2(S) is a frame for L2(S). However, when ab > 1 and S ≠ ℝ, 𝒢(g, a, b) cannot be a frame in L2(ℝ) for any g ∈ L2(ℝ), while it is possible that there exists some g such that 𝒢(g, a, b) is a frame for L2(S). So the projections of Gabor frames in L2(ℝ) onto L2(S) cannot cover all Gabor frames in L2(S). This paper considers Gabor systems in L2(S). In order to use the Zak transform, we only consider the case where the product ab is a rational number. With the help of a suitable Zak transform matrix, we characterize Gabor frames for L2(S) of the form 𝒢(g, a, b), and obtain an expression for the canonical dual of a Gabor frame. We also characterize the uniqueness of Gabor duals of type I (respectively, type II).

FRÉCHET DIFFERENTIABILITY OF THE NORM IN A SOBOLEV SPACE WITH A VARIABLE EXPONENT

Let Ω be a domain in ℝN, let [Formula: see text] be such that p(x) > 1 for all [Formula: see text], let W1,p(⋅) (Ω) be the Sobolev space with variable exponent p(⋅), let Γ0 be a dΓ-measurable subset of Γ = ∂Ω that satisfies dΓ-meas Γ0 > 0, and let UΓ0 = {u ∈ W1,p(⋅)(Ω); tr u = 0 on Γ0}. It is shown that the map u ∈ UΓ0 ↦ ‖u‖0,p(⋅), ∇ = ‖|∇u|‖0,p(⋅) is a Fréchet-differentiable norm on UΓ0, and a formula expressing the Fréchet derivative of this norm at any nonzero u ∈ UΓ0 is given. We also show that, if p(x) ≥ 2 for all [Formula: see text], (UΓ0, ‖u‖0,p(⋅), ∇) is uniformly convex. Using properties of duality mappings defined on Banach spaces having a Fréchet-differentiable norm, we give the explicit form of continuous linear functionals on (UΓ0, ‖u‖0,p(⋅), ∇). It is also shown that the space UΓ0 and its dual have the same Krein–Krasnoselski–Milman dimension.