The Poisson Integral of a Singular Measure

1983 ◽  
Vol 35 (4) ◽  
pp. 735-749 ◽  
Author(s):  
Patrick Ahern
Keyword(s):  
Lower Bound ◽  
Euclidean Space ◽  
Lebesgue Measure ◽  
Open Subset ◽  
Borel Measure ◽  
The Other ◽  
Poisson Integral ◽  
Singular Measure ◽  
Almost Everywhere ◽  
Image Position

Let σ be a finite positive singular Borel measure defined on Euclidean space RN. For w ∈ RN and y > 0, its Poisson integral is defined by the formulawhere CN is chosen so thatSince σ is singular, almost everywhere with respect to Lebesgue measure on RN. On the other hand, almost everywhere dσ. It follows that for all sufficiently small y,is a non-empty open subset of RN. If σ has compact support then |Ey| → 0 as y → 0, where |Ey| denotes the Lebesgue measure of Ey. In this paper we give a lower bound on the rate at which |Ey| may go to zero. The lower bound depends on the smoothness of the measure; the smoother the measure, the more slowly |Ey| may approach 0.

Additive functions of intervals and Hausdorff measure

1946 ◽  
Vol 42 (1) ◽  
pp. 15-23 ◽  
Author(s):  
P. A. P. Moran
Keyword(s):  
Lower Bound ◽  
Euclidean Space ◽  
Hausdorff Measure ◽  
Convex Sets ◽  
Infinite Sequence ◽  
Small Positive Number ◽  
Additive Functions ◽  
Image Position ◽  
Bounded Sets ◽  
Increasing Function

Consider bounded sets of points in a Euclidean space Rq of q dimensions. Let h(t) be a continuous increasing function, positive for t>0, and such that h(0) = 0. Then the Hausdroff measure h–mE of a set E in Rq, relative to the function h(t), is defined as follows. Let ε be a small positive number and suppose E is covered by a finite or enumerably infinite sequence of convex sets {Ui} (open or closed) of diameters di less than or equal to ε. Write h–mεE = greatest lower bound for any such sequence {Ui}. Then h–mεE is non-decreasing as ε tends to zero. We define

Saturated systems of symmetric convex domains; results of Eggleston, Bambah and Woods

1973 ◽  
Vol 74 (1) ◽  
pp. 107-116 ◽  
Author(s):  
Vishwa Chander Dumir ◽  
Dharam Singh Khassa
Keyword(s):  
Lower Bound ◽  
Lebesgue Measure ◽  
Convex Domain ◽  
Gauge Function ◽  
Symmetric Convex ◽  
Convex Domains ◽  
Strictly Convex ◽  
Point Set ◽  
Image Position

Let K be a closed, bounded, symmetric convex domain with centre at the origin O and gauge function F(x). By a homothetic translate of K with centre a and radius r we mean the set {x: F(x−a) ≤ r}. A family ℳ of homothetic translates of K is called a saturated family or a saturated system if (i) the infimum r of the radii of sets in ℳ is positive and (ii) every homothetic translate of K of radius r intersects some member of ℳ. For a saturated family ℳ of homothetic translates of K, let S denote the point-set union of the interiors of members of ℳ and S(l), the set S ∪ {x: F(x) ≤ l}. The lower density ρℳ(K) of the saturated system ℳ is defined bywhere V(S(l)) denotes the Lebesgue measure of the set S(l). The problem is to find the greatest lower bound ρK of ρℳ(K) over all saturated systems ℳ of homothetic translates of K. In case K is a circle, Fejes Tóth(9) conjectured thatwhere ϑ(K) denotes the density of the thinnest coverings of the plane by translates of K. In part I, we state results already known in this direction. In part II, we prove that ρK = (¼) ϑ(K) when K is strictly convex and in part III, we prove that ρK = (¼) ϑ(K) for all symmetric convex domains.

Mass transference principle for limsup sets generated by rectangles

2015 ◽  
Vol 158 (3) ◽  
pp. 419-437 ◽  
Author(s):  
BAO-WEI WANG ◽  
JUN WU ◽  
JIAN XU
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Lebesgue Measure ◽  
Hausdorff Measure ◽  
Unit Cube ◽  
Transference Principle ◽  
Theoretical Statement ◽  
Formula Group ◽  
Image Position ◽  
Full Lebesgue Measure

AbstractWe generalise the mass transference principle established by Beresnevich and Velani to limsup sets generated by rectangles. More precisely, let {xn}n⩾1 be a sequence of points in the unit cube [0, 1]d with d ⩾ 1 and {rn}n⩾1 a sequence of positive numbers tending to zero. Under the assumption of full Lebesgue measure theoretical statement of the set \begin{equation*}\big\{x\in [0,1]^d: x\in B(x_n,r_n), \ {{\rm for}\, {\rm infinitely}\, {\rm many}}\ n\in \mathbb{N}\big\},\end{equation*} we determine the lower bound of the Hausdorff dimension and Hausdorff measure of the set \begin{equation*}\big\{x\in [0,1]^d: x\in B^{a}(x_n,r_n), \ {{\rm for}\, {\rm infinitely}\, {\rm many}}\ n\in \mathbb{N}\big\},\end{equation*} where a = (a1, . . ., ad) with 1 ⩽ a1 ⩽ a2 ⩽ . . . ⩽ ad and Ba(x, r) denotes a rectangle with center x and side-length (ra1, ra2,. . .,rad). When a1 = a2 = . . . = ad, the result is included in the setting considered by Beresnevich and Velani.

Continuity of solutions of Schrödinger equations

1991 ◽  
Vol 110 (3) ◽  
pp. 581-597
Author(s):  
Mitsuru Nakai
Keyword(s):  
Euclidean Space ◽  
Total Variation ◽  
Open Subset ◽  
Radon Measure ◽  
Dimensional Euclidean Space ◽  
Schrödinger Equations ◽  
Kato Class ◽  
Continuity Of Solutions ◽  
Image Position ◽  
Newtonian Kernel

We denote by N(x, y) the Newtonian kernel on the d-dimensional Euclidean space (where d ≥ 2) so that N(x, y) = log|x–y|-1 for d = 2 and N(x, y) = |x−y|2−d for d ≥ 3. A signed Radon measure μ on an open subset Ω in d is said to be of Kato class iffor every y in Ω. where |μ| is the total variation measure of μ.

Some Lemmas on Interpolating Blaschke Products and a Correction

1969 ◽  
Vol 21 ◽  
pp. 531-534 ◽  
Author(s):  
A. Kerr-Lawson
Keyword(s):  
Blaschke Product ◽  
Unit Disc ◽  
Inner Function ◽  
Open Unit ◽  
Inner Functions ◽  
Singular Measure ◽  
Radial Limits ◽  
Almost Everywhere ◽  
Image Position ◽  
Interpolating Blaschke Products

A Blaschke product on the unit disc,where |c|= 1 and kis a non-negative integer, is said to be interpolatingif the conditionCis satisfied for a constant δ independent of m.A Blaschke product always belongs to the set I of inner functions; it has norm 1 and radial limits of modulus 1 almost everywhere. The most general inner function can be uniquely factored into a product BS,where Bis a Blaschke product andfor some positive singular measure μ(θ) on the unit circle. The discussion will be carried out in terms of the hyperbolic geometry on the open unit disc D,its metricand its neighbourhoods N(x, ∈) = ﹛z′ ∈ D: Ψ(z, z′) < ∈ ﹜

scholarly journals On Fourier transform multipliers in Lp

1972 ◽  
Vol 13 (2) ◽  
pp. 219-223
Author(s):  
G. O. Okikiolu
Keyword(s):  
Fourier Transform ◽  
Euclidean Space ◽  
Characteristic Function ◽  
Lebesgue Measure ◽  
Real Numbers ◽  
Measurable Functions ◽  
Image Position

We denote by R the set of real numbers, and by Rn, n ≧ 2, the Euclidean space of dimension n. Given any subset E of Rn, n ≧ 1, we denote the characteristic function of E by xE, so that XE(x) = 0 if x ∈ E; and XE(X) = 0 if x ∈ Rn/E.The space L(Rn) Lp consists of those measurable functions f on Rn such that is finite. Also, L∞ represents the space of essentially bounded measurable functions with ║f║>0; m({x: |f(x)| > x}) = O}, where m represents the Lebesgue measure on Rn The numbers p and p′ will be connected by l/p+ l/p′= 1.

scholarly journals A Note on the Kawada-into Theorem

1963 ◽  
Vol 13 (4) ◽  
pp. 295-296 ◽  
Author(s):  
John S. Pym
Keyword(s):  
Compact Group ◽  
Haar Measure ◽  
Borel Measure ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Almost Everywhere ◽  
Regular Borel Measure ◽  
Image Position ◽  
Complex Valued

If µ is a bounded regular Borel measure on a locally compact group G, and L1(G) denotes the class of complex-valued functions which are integrable with respect to the left Haar measure m of G, then, for each f∈L1(G),defines almost everywhere (a.e.) with respect to m a function μ*f which is again in L1(G). The measure μ will be called isotone on G mapping f→μ*f is isotone, i.e. f≧0 a.e. (m) if and only if μ*f≧0 a.e. (m).

scholarly journals Multipliers of Bergman spaces into Lebesgue spaces

1986 ◽  
Vol 29 (1) ◽  
pp. 125-131 ◽  
Author(s):  
Daniel H. Luecking
Keyword(s):  
Lebesgue Measure ◽  
Unit Disk ◽  
Analytic Functions ◽  
Lebesgue Space ◽  
Borel Measure ◽  
Bergman Spaces ◽  
Open Unit ◽  
Open Unit Disk ◽  
Lebesgue Spaces ◽  
Image Position

Let U be the open unit disk in the complex plane endowed with normalized Lebesgue measure m. will denote the usual Lebesgue space with respect to m, with 0<p<+∞. The Bergman space consisting of the analytic functions in will be denoted . Let μ be some positivefinite Borel measure on U. It has been known for some time (see [6] and [9]) what conditions on μ are equivalent to the estimate: There is a constant C such thatprovided 0<p≦q.

CONVERGENCE OF THE CONTINUOUS WAVELET TRANSFORMS ON THE ENTIRE LEBESGUE SET OF Lp-FUNCTIONS

2011 ◽  
Vol 09 (04) ◽  
pp. 675-683 ◽  
Author(s):  
RAVSHAN ASHUROV
Keyword(s):  
Lebesgue Measure ◽  
Wavelet Transforms ◽  
Measure Zero ◽  
The Other ◽  
Continuous Wavelet ◽  
Exceptional Set ◽  
Localization Principle ◽  
Continuous Wavelet Transforms ◽  
Almost Everywhere ◽  
Lebesgue Set

The almost everywhere convergence of wavelets transforms of Lp-functions under minimal conditions on wavelets is well known. But this result does not provide any information about the exceptional set (of Lebesgue measure zero), where convergence does not hold. In this paper, under slightly stronger conditions on wavelets, we prove convergence of wavelet transforms everywhere on the entire Lebesgue set of Lp-functions. On the other hand, practically all the wavelets, including Haar and "French hat" wavelets, used frequently in applications, satisfy our conditions. We also prove that the same conditions on wavelets guarantee the Riemann localization principle in L1 for the wavelet transforms.

A harmonic quadrature formula characterizing open strips

1993 ◽  
Vol 113 (1) ◽  
pp. 147-151 ◽  
Author(s):  
D. H. Armitage ◽  
C. S. Nelson
Keyword(s):  
Harmonic Function ◽  
Lebesgue Measure ◽  
Quadrature Formula ◽  
Open Subset ◽  
Harmonic Functions ◽  
Mean Value ◽  
Open Ball ◽  
Mean Value Property ◽  
Image Position ◽  
Dimensional Lebesgue Measure

Let γn denote n-dimensional Lebesgue measure. It follows easily from the well-known volume mean value property of harmonic functions that if h is an integrable harmonic function on an open ball B of centre ξ0 in ℝn, where n ≥ 2, thenA converse of this result is due to Kuran [8]: if D is an open subset of ℝn such that γn(D) < + ∞ and if there exists a point ξo∈D such thatfor every integrable harmonic function h on D, then D is a ball of centre ξ0. Armitage and Goldstein [2], theorem 1, showed that the same conclusion holds under the weaker hypothesis that (1·2) holds for all positive integrable harmonic functions h on D.

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