Differentiable Functions on Bad Domains by Vladimir G Maz'ya, Sergei Poborchi

t if>(z) , t Eras z tends to t E r along any nontangential way.

Let A12 = 0 and let the operator All (or A 22 ) be Fredholm in X. 16) and where m = dim (A21 (ker All)/Im A22 nA21(ker All)). We also formulate the following result for the same case n [161]). 37. Let All be Fredholm in X and let Rll be its regularizer. Then A is Fredholm in X 2 if and only if the operator A22 - A21RllA12 is Fredholm in X and Corollary. Let All be Fredholm in X and quasi commute either with A12 or A 21 . Then A is Fredholm in X2 if and only if the operator AllA22 - A21A12 is Fredholm in X and 13 1.

L(r) is the continuous step-type function, equal to 1 when r :S r2, to ~1;~~) when rk :S r :S rk + 1, and to when rk + 1 :S r :S rk+l, where the numbers rk are defined by the condition that la(x)1 :S for Ixl ~ rk, k = 1,2, .... i i 3. On Fredholmness of convolution type operators 39 5) In the case n = 1, it suffices to note that functions a(x) E BV(-oo, oo) have the limits lim a( x); see Dunford and Schwartz [50], p. 171. 30. Any convolution operator h BO'es(Rn) into Co(Rn). p with h(x) E L 1(Rn) maps We refer the reader for the proof of this lemma to Karapetiants and Samko [110].

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