On the m-fold product of fractional operators

In this work, using the m-fold product of fractional integral and maximal operators, we prove that the boundedness of these fractional operators and their corresponding multilinear fractional operators under some conditions on weighted variable exponent Lorentz spaces.Publisher's Versio

Boundedness of the Bilinear Littlewood-Paley Square Function on Variable Lorentz Spaces

International Conference on Numerical Analysis and Applied Mathematics (ICNAAM) -- SEP 19-25, 2016 -- Rhodes, GREECEWOS: 000410159800302Let m is an element of S (R). For n is an element of Z define m(n) (zeta) = m(zeta- n) and let S-n be the bilinear multiplier operator associated with m(n). In this paper, if p (infinity) = r (infinity), p (0) = r (0), 1/p(t) + 1/s(t) >= 1/q(t) + 1/r(t), r (t) <= s (t), 1/s(t) = 1/s(1)(t) + 1/s(2)(t) and 2/r(t) + 1 = 2/r(1)(t) + 2/r(2)(t), then it is proved that the bilinear Littlewood-Paley square function S (f, g) is bounded on from L-r1(.),L- s1(.) (R) x L-r2(.),L- s2(.) (R) to L-p(.),L- q(.) (R)

Bilinear Multipliers of Small Lebesgue Spaces

Let G be a compact abelian metric group with Haar measure lambda and (G) over cap its dual with Haar measure mu. Assume that 1 = 0. Let L-(pi' ,L-theta (G), (i = 1, 2, 3) be small Lebesgue spaces. A bounded sequence m(xi, eta) defined on G (over cap) x G (over cap) is said to be a bilinear multiplier on G of type [(p'(1); (p'(2); (p'(3)]. if the bilinear operator B-m associated with the symbol m B-m (f, g) (x) = Sigma(delta is an element of G)Sigma(t is an element of G) (f) over cap (s) (g) over cap (t) m(s, t) (s + t, x) defines a bounded bilinear operator from L-(p'1,L- theta (G) x L-(p2',L-theta (G) into L-(p3',L-theta (G). We denote by BM theta [(p(1)' ; (p(2)' ; (p(3)'] the space of all bilinear multipliers of type [(p(1)'; (p(2)'; (p(3)'](theta). In this paper, we discuss some basic properties of the space BM. [(p(1)'; (p(2)'; (p(3)'] and give examples of bilinear multipliers

On Function Spaces With Wavelet Transform In Lp ω Rd × R+

Let ?1 and ?2 be weight functions on R d , R d × R+, respectively. Throughout this paper, we define D p,q ?1,?2 R d to be the vector space of f ? L p ?1 R d such that the wavelet transform Wgf belongs to L q ?2 R d × R+ for 1 ? p,q < ?, where 0 6= g ? S R d. We endow this space with a sum norm and show that Dp,q ?1,?2 R d becomes a Banach space. We discuss inclusion properties, and compact embeddings between these spaces and the dual of D p,q ?1,?2 R d . Later we accept that the variable s in the space D p,q ?1,?2 R d is fixed. We denote this space by D p,q ?1,?2 s R d, and show that under suitable conditions Dp,q ?1,?2 s R d is an essential Banach Module over L1 ?1 R d. We obtain its approximate identities. At the end of this work we discuss the multipliers from D p,q ?1,?2 s R d into L??-11Rd , and from L 1 ?1 R d into D p,q ?1,?2 s R d

Bilinear multipliers of weighted Lebesgue spaces and variable exponent Lebesgue spaces

Let 1 < p1,p2 < to, 0 < p3 <to and c1, c2, c3 be weight functions on R". Assume that c1 , c2 are slowly increasing functions. We say that a bounded function m(£, n) defined on R" x R" is a bilinear multiplier on R" of type (p1, coy,p2, c2;p3, c3) (shortly (co1, c2, c3)) if 2n ¡{i~+nx) Bm(f,g)(x) = í Í f(Ş)g(n)m(Ş,n)e2n!{Ş+nx) dŞ dn R" R" is a bounded bilinear operator from L^ (R") x LP22(R") to LpJ3(R"). We denote by BM(p1,M1;p2,M2;p3,M3) (shortly BM(m1,m2,m3)) the vector space of bilinear multipliers of type (m1,m2,m3). In this paper first we discuss some properties of the space BM(m1,m2,m3). Furthermore, we give some examples of bilinear multipliers. At the end of this paper, by using variable exponent Lebesgue spaces LPl(x)(R") Lp2M(R") and LP3(x)(R"), we define the space of bilinear multipliers from Lpl(x)(R") x LP2(x)(R") to LP3(x)(R") and discuss some properties of this space

Bilinear Multipliers Of Weighted Wiener Amalgam Spaces And Variable Exponent Wiener Amalgam Spaces

Let ?1, ?2 be slowly increasing weight functions, and let ?3 be any weight function on Rn. Assume that m(?,?) is a bounded, measurable function on Rn×Rn. We define Bm(f,g)(x)=?Rn?Rnfˆ(?)gˆ(?)m(?,?)e2?i??+?,x?d?d? for all f,g?C?c(Rn). We say that m(?,?) is a bilinear multiplier on Rn of type (W(p1,q1,?1;p2,q2,?2;p3,q3,?3)) if Bm is a bounded operator from W(Lp1,Lq1?1)×W(Lp2,Lq2?2) to W(Lp3,Lq3?3), where 1?p1?q1<?, 1?p2?q2<?, 1<p3,q3??. We denote by BM(W(p1,q1,?1;p2,q2,?2;p3,q3,?3)) the vector space of bilinear multipliers of type (W(p1,q1,?1;p2,q2,?2;p3,q3,?3)). In the first section of this work, we investigate some properties of this space and we give some examples of these bilinear multipliers. In the second section, by using variable exponent Wiener amalgam spaces, we define the bilinear multipliers of type (W(p1(x),q1,?1;p2(x),q2,?2;p3(x),q3,?3)) from W(Lp1(x),Lq1?1)×W(Lp2(x),Lq2?2) to W(Lp3(x),Lq3?3), where p*1,p*2,p*3<?, p1(x)?q1, p2(x)?q2, 1?q3?? for all p1(x),p2(x),p3(x)?P(Rn). We denote by BM(W(p1(x),q1,?1;p2(x),q2,?2;p3(x),q3,?3)) the vector space of bilinear multipliers of type (W(p1(x),q1,?1;p2(x),q2,?2;p3(x),q3,?3)). Similarly, we discuss some properties of this space

The spaces of bilinear multipliers of weighted Lorentz type modulation spaces

#nofulltext# --- Gürkanlı, Ahmet Turan (Arel Author)Fix a nonzero window g?S(Rn), a weight function w on R2n and 1?p,q??. The weighted Lorentz type modulation space M(p,q,w)(Rn) consists of all tempered distributions f?S'(Rn) such that the short time Fourier transform Vgf is in the weighted Lorentz space L(p,q,wdµ)(R2n). The norm on M(p,q,w)(Rn) is ?f?M(p,q,w)=?Vgf?pq,w. This space was firstly defined and some of its properties were investigated for the unweighted case by Gürkanlı in [9] and generalized to the weighted case by Sandıkçı and Gürkanlı in [16]. Let 1<p1,p2<?, 1?q1,q2<?, 1?p3,q3??, ?1,?2 be polynomial weights and ?3 be a weight function on R2n. In the present paper, we define the bilinear multiplier operator from M(p1,q1,?1)(Rn)×M(p2,q2,?2)(Rn) to M(p3,q3,?3)(Rn) in the following way. Assume that m(?,?) is a bounded function on R2n, and define Bm(f,g)(x)=?Rn?Rnf^(?)g^(?)m(?,?)e2?i??+?,x?d?d? for allf,g?S(Rn). The function m is said to be a bilinear multiplier on Rn of type (p1,q1,?1;p2,q2,?2;p3,q3,?3) if Bm is the bounded bilinear operator from M(p1,q1,?1)(Rn)×M(p2,q2,?2)(Rn) to M(p3,q3,?3)(Rn). We denote by BM(p1,q1,?1;p2,q2,?2)(Rn) the space of all bilinear multipliers of type (p1,q1,?1;p2,q2,?2;p3,q3,?3), and define ?m?(p1,q1,?1;p2,q2,?2;p3,q3,?3)=?Bm?. We discuss the necessary and sufficient conditions for Bm to be bounded. We investigate the properties of this space and we give some examples

Spectrum of Zariski Topology in Multiplication Krasner Hypermodules

In this paper, we define the concept of pseudo-prime subhypermodules of hypermodules as a generalization of the prime hyperideal of commutative hyperrings. In particular, we examine the spectrum of the Zariski topology, which we built on the element of the pseudo-prime subhypermodules of a class of hypermodules. Moreover, we provide some relevant properties of the hypermodule in this topological hyperspace

Electrophysiologic evaluation of facial nerve functions after oxaliplatin treatment

Purpose: This study analyzes the effect of oxaliplatin treatment on the facial nerve. The facial nerve is the most commonly paralyzed cranial motor nerve because it advances through a long, curved bone canal. Electroneurography and blink reflex are the electrophysiological measurements used for evaluating facial nerve function. Oxaliplatin is a cytotoxic agent used in adjuvant or palliative systemic therapy for colorectal cancer treatment. Methods: This study was performed on 20 individuals who were at least 18 years old at Hacettepe University Ear Nose Throat Department, Audiology and Speech Disorders Unit, and Neurology Division EMG Laboratory as they received oxaliplatin treatment from Hacettepe University Oncology Hospital. Electroneurography and blink-reflex values were recorded and examined. The parameters taken during the second and fourth months were compared for this purpose. Results: This study shows that the prolongation of distal latencies of compound muscle action potential is statistically significant, the amplitudes showed no difference. The ENoG results were analyzed, the prolongation of latency measurements between pre-treatment and the fourth month after treatment were statistically significant. The blink-reflex results showed that comparison with the baseline values, the prolongation of latencies in R1 measurements between pre-treatment, the second month, and the fourth month were significant. Conclusions: The facial nerve is affected asymptomatically by oxaliplatin treatment. During oxaliplatin treatment, the evaluation of facial nerve function could be beneficial for patients by improving their quality of life. Electroneurography and blink-reflex tests can be used in the early evaluations of different medicines to determine their neurotoxicity