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## Singular Integrals on Product Homogeneous Groups

### Integral Equations and Operator Theory (2013-05-01) 76: 55-79 , May 01, 2013

We consider singular integral operators with rough kernels on the product space of homogeneous groups. We prove *L*^{p} boundedness of them for
$${p \in (1,\infty)}$$
under a sharp integrability condition of the kernels.

## Weak (1,1) Estimates for Littlewood-Paley Functions with Rough Kernels

### Proceedings of the Second ISAAC Congress (2000-01-01) 7: 83-87 , January 01, 2000

We consider Marcinkiewicz integrals on *R*^{n} and *T*^{
n}
arising from rough kernels satisfying the *L* log *L*-condition on *S*^{n}^{-1} and prove the weak type (1,1) estimates. We also prove the weighted weak type (1,1) estimates on *R*^{n}
with *A*_{1}-weights. In this case the *L* log *L*-condition is replaced by the *L*^{q}-condition with *q* > 1.

## Estimates for Littlewood-Paley Functions and Extrapolation

### Integral Equations and Operator Theory (2008-11-01) 62: 429-440 , November 01, 2008

### Abstract.

We prove certain *L*^{p}-estimates for Littlewood-Paley functions arising from rough kernels. The estimates are useful for extrapolation to prove *L*^{p}-boundedness of the Littlewood-Paley functions under a sharp kernel condition.

## Weak type estimates for some maximal operators on the weighted Hardy spaces

### Arkiv för Matematik (1995-10-01) 33: 377-384 , October 01, 1995

Weighted weak type estimates are proved for some maximal operators on the weighted Hardy spaces*H*_{ω}^{p}
(0 <*p* < 1, ω ∈*A*_{1}) (0<p<1, ω∞A_{1}); in particular, weighted weak type endpoint estimates are obtained for the maximal operators arising from the Bochner-Riesz means and the spherical means on*H*_{ω}^{p}
.

## Spherical square functions of Marcinkiewicz type with Riesz potentials

### Archiv der Mathematik (2017-03-08): 1-12 , March 08, 2017

We prove a pointwise equivalence between a spherical square function composed with the Riesz potential and a Littlewood–Paley function arising from the Bochner–Riesz operators. Also, its application to the theory of Sobolev spaces will be given.

## Weighted weak type (1,1) estimates for singular integrals with non-isotropic homogeneity

### Arkiv för Matematik (2016-04-01) 54: 157-180 , April 01, 2016

We prove sharp weighted weak type (1,1) estimates for rough singular integral operators on homogeneous groups. Similar results are shown for singular integrals on $\mathbb{R}^{2}$ with the generalized homogeneity.

## Fibonacci Sequence and its Generalizations Hidden in Algorithms for Generating Morse Codes

### Applications of Fibonacci Numbers (1993-01-01): 481-486 , January 01, 1993

In such advanced devices as computers, facsimiles and CD players, information is sent by digital signals. The Morse code, composed of dots and dashes, is a forerunner of digital communications codes. The code was originated about 1840 by the American inventor Samuel F. B. Morse and his assistant Alfred Vail. In 1851 an international convention improved the original Morse’s code and the resulting newer code is called the International Morse code.

## On Matrix Representations of Generalized Fibonacci Numbers and Their Applications

### Applications of Fibonacci Numbers (1993-01-01): 487-496 , January 01, 1993

Many kinds of generalizations of Fibonacci numbers (e.g., see [2], [4], [7], [13], [15]) have been investigated and many interesting properties of the generalized Fibonacci numbers have been obtained. Furthermore, some notable generalizations of the Lucas Number (e.g., see [3], [6], [8]) have been undertaken.