Let G be a locally compact group, w be a continuous weight function on G, and L^1(G,w) be the corresponding Beurling algebra. In this thesis, we study weak amenability of L^1(G,w) and of its centre ZL^1(G,w) for non-commutative locally compact groups G. We first give examples to show that the condition that characterizes weak amenability of L^1(G,w) for commutative groups G is no longer sufficient for the non-commutative case. However, we prove that this condition remains necessary for all [IN] groups G. We also provide a necessary condition for weak amenability of L^1(G,w) of a different nature, which, among other things, allows us to obtain a number of significant results on weak amenability of l^1(F_2,w) and l^1((ax+b),w). We then study the relation between weak amenability of the algebra L^1(G,w) on a locally compact group G and the algebra L^1(G/H,^w) on the quotient group G/H of G over a closed normal subgroup H with an appropriate weight ^w induced from w. We give an example showing that L^1(G,w) may not be weakly amenable even if both L^1(G/H,^w) and L^1(H,w|_H) are weakly amenable. On the other hand, by means of constructing a generalized Bruhat function on G, we establish a sufficient condition under which weak amenability of L^1(G,w) implies that of L^1(G/H,^w). In particular, with this approach, we prove that weak amenability of the tensor product of L^1(G_1,w_1) and L^1(G_2,w_2) implies weak amenability of both Beurling algebras L^1(G_1,w_1) and L^1(G_2,w_2), provided the weights w_1, w_2 are bounded away from zero. However, given a general weight on the direct product G of G_1 and G_2, weak amenability of L^1(G,w) usually does not imply that of L^1(G_1,w|_{G_1}), even if both G_1, G_2 are commutative. We provide an example to illustrate this. While studying the centres ZL^1(G,w) of L^1(G,w), we characterize weak amenability of ZL^1(G,w) for connected [SIN] groups G, establish a necessary condition for weak amenability of ZL^1(G,w) in the case when G is an [FC] group, and give a sufficient condition for the case when G is an [FD] group. In particular, we obtain some positive results on weak amenability of ZL^1(G,w) for a compactly generated [FC] group G with a polynomial weight w. Finally, we briefly discuss the derivation problem for weighted group algebras and present a partial solution to it.

<p>We use common notation &lowast; for distribution (Scshwartz), (M_p) (Beurling) i {M_p} (Roumieu) setting. We introduce and study new (ultra) distribution spaces, the test function spaces&nbsp;<em>D^&lowast;_E</em>&nbsp; and their strong duals <em>D^{<span style="font-size: 10px;">&#39;</span>&lowast;}_E&rsquo;*</em>.These spaces generalize the spaces <em>D^&lowast;_L^q , D&#39;^&lowast;_L^p , B&rsquo;*</em>&nbsp;and their weighted versions. The construction of our new (ultra)distribution &nbsp;spaces is based on the analysis of a suitable translation-invariant Banach space of (ultra)distribution <em>E</em>&nbsp;with continuous translation group, which turns out to be a convolution module over the Beurling algebra&nbsp;<em>L¹_&omega;</em>, where the weight &nbsp;&omega; is related to the translation operators on <em>E</em>.&nbsp;The&nbsp;Banach space&nbsp;<em>E</em>^&rsquo;_&lowast;&nbsp;stands for&nbsp;<em>L¹_&omega;ˇ &lowast; E</em>&rsquo;.&nbsp;We apply our results to the study of the&nbsp;convolution of ultradistributions. The spaces of convolutors&nbsp;<em>O<span style="font-size: 12px;">&rsquo;^&lowast;</span><span style="font-size: 8.33333px;">C</span></em><span style="font-size: 12px;"><em>&nbsp;(</em><strong>R</strong><em>ⁿ)</em>&nbsp;</span>for tempered&nbsp;ultradistributions are analyzed via the duality with respect to the test function<br />spaces<span style="font-size: 12px;">&nbsp;<em>O^&lowast;_C (</em><strong>R</strong><em>ⁿ)</em>,&nbsp;</span>introduced in this thesis. Using the properties of translationinvariant<br />Banach space of ultradistributions <em>E</em> we obtain a full characterization of<br />the general convolution of Roumieu ultradistributions via the space of integrable<br />ultradistributions is obtained. We show: The convolution of two Roumieu ultradistributions&nbsp;<span style="font-size: 12px;"><em>T, S &isin; D&rsquo;^{Mp} (</em><strong>R</strong><em>ⁿ)&nbsp;</em> exists if and only if&nbsp;<em>(</em></span><em>&phi;</em><span style="font-size: 12px;"><em>&nbsp;&lowast; &Scaron;) T &isin; D^&rsquo;{Mp}_L¹(</em><strong>R</strong><em>ⁿ)</em>&nbsp; for every </span><em>&phi;</em><span style="font-size: 12px;"><em>&nbsp;&isin; D ^{Mp} (</em><strong>R</strong><em>ⁿ)</em>.&nbsp;</span>We study boundary values of holomorphic functions defined in tube domains. New edge of the wedge theorems are obtained. The results<br />are then applied to represent<span style="font-size: 12px;">&nbsp;<em>D&rsquo;_E&rsquo;*</em></span><span style="font-size: 12px;">&nbsp;&nbsp;</span>as a quotient space of holomorphic functions.<br />We also give representations of elements of<span style="font-size: 12px;">&nbsp;<em>D&rsquo;_E&rsquo;*</em></span><span style="font-size: 12px;">&nbsp;&nbsp;</span>via the heat kernel method.</p> / <p>Koristimo oznaku &lowast; za distribuciono (Svarcovo), (Mp) (Berlingovo) i&nbsp;{Mp} (Roumieuovo) okruženje. Uvodimo i prouavamo nove (ultra)distribucione&nbsp;prostore, &nbsp;test funkcijske prostore <em>D</em>^&lowast;_E i njihove duale <em>D^&#39;</em>^&lowast;_{<em>E&#39;*</em>}.&nbsp;&nbsp;Ovi prostori uop&scaron;tavaju&nbsp;<br />prostore <em>D</em>^&lowast;_Lq , <em>D</em>^{&#39;&lowast;}_Lp , <em>B^&#39;</em>^&lowast; i njihove težinske verzije. Konstrukcija na&scaron;ih novih&nbsp;<br />(ultra)distribucionih prostora je zasnovana na analizi odgovarajuićh translaciono&nbsp;<br />- invarijantnih Banahovih prostora (ultra)distribucija koje označavamo sa&nbsp;<em>E</em>. Ovi prostori imaju neprekidnu grupu translacija, koja je konvolucioni modul&nbsp;nad &nbsp;Beurlingovom algebrom L¹_&omega;, gde je težina &omega; povezana sa operatorima translacije&nbsp;<br />prostora <em>E</em>. Banahov prostor <em>E^&#39;</em>_{&lowast;&nbsp;}označava prostor <em>L</em>¹_&omega;˅ &lowast; <em>E^&#39;</em>. Koristeći dobijene&nbsp;<br />rezultata proučavamo konvoluciju ultradistribucija. Prostori konvolutora &nbsp;<em>O^&#39;</em>^&lowast;_{<em>C&nbsp;</em>}(<strong>R</strong>ⁿ)&nbsp;temperiranih ultradistribucija, analizirani su pomoću dualnosti&nbsp;<br />test funkcijskih prostora <em>O</em>^&lowast;_<em>C</em> (<strong>R</strong>ⁿ), definisanih u ovoj tezi. Koristeći svojstva&nbsp;<br />translaciono - invarijantnih Banahovih prostora temperiranih ultradistribucija,&nbsp;<br />opet označenih sa <em>E</em>, dobijamo karakterizaciju konvolucije Romuieu-ovih &nbsp;ultradistribucija,&nbsp;preko integrabilnih ultradistribucija. Dokazujemo da: konvolucija&nbsp;<br />dve Roumieu-ove ultradistribucija <em>T</em>, <em>S</em> &isin; <em>D^&#39;</em>^{Mp}&nbsp;(<strong>R</strong>ⁿ) postoji ako i samo ako (&phi; &lowast; <em>S</em>ˇ)<em>T</em> &isin; <em>D^&#39;</em>^{Mp}&nbsp;_L¹ (<strong>R</strong>ⁿ) za svaki &phi; &isin; <em>D</em>^{Mp}(<strong>R</strong>ⁿ). Takođe, proučavamo granične vrednosti holomorfnih funkcija definisanih na tubama. Dokazane su nove teoreme &rdquo;otrog klina&rdquo;. Rezultati se zatim koriste za prezentaciju <em>D^&#39;_E^&#39;</em>_{&lowast;&nbsp;}preko faktor prostora holomorfnih funkcija. Takođe, data je prezentacija elemente <em>D</em>^&#39;_{<em>E^&#39;</em>&lowast;&nbsp;}koristeći heat kernel metode.</p>