Boundedness Of Bilinear Fractional Integrals (Fractional Integrals Along Non-Homogeneous Curves) On Vanishing Generalized Morrey Spaces (Lebesgue Spaces)

This thesis is divided into two parts:(i)boundedness of bilinear fractional integrals on vanishing Morrey spaces;(ii)boundedness of fractional integrals along curves on Lebesgue spaces.On one hand,the author establishes the boundedness of the bilinear fractional in-tegral operator Ba and the subbilinear fractional maximal operator Ma on vanishing generalized Morrey spaces V0Lp,?(Rn),V?Lp,?(Rn)and V(*)Lp,?(Rn).To this end,the author first obtains a significant estimate about mq,?(B?(f,g);x,r)by using the product of two classical fractional integral operators I? to control Ba.Then the au-thor obtains the boundedness of B? on V0Lp,?(Rn)via dividing mq,?(B?(f,g);x,r)into four parts and estimating them separately.The boundedness of Ba on V?Lp,?(Rn)can be proved by some similar methods as the proof of the boundedness of Ba on V0Lp,?(Rn).Next the author shows that Ba can be controlled by subbilinear maxi-mal operator M and Ma' with ?'>? via dividing the region of the integral about Ba into countable rings.By this,the Holder inequality and the boundedness of M on V(*)Lp,?(Rn),the author establishes the boundedness of Ba on V(*)Lp,?(Rn).Further-more,the boundedness of Ma on the above three vanishing Morrey subspaces is an easy consequence of the fact that Ma can be controlled pointwisely by B?.Finally,the author also gives some specific examples of the main results of this part.On the other hand,the author shows that the boundedness of the bilinear fractional integrals I?1,?2,?3 along curves(t?1,at?2+b)with ?2>?1?1,a?(?2/?1,?)and b?R on Lebesgue spaces,which extends the results of bilinear fractional integrals along homogeneous curves of Junfeng Li and Peng Liu.Via changes of variables,the author reduces the proof to the boundedness of the operator T?,?.To this end,the author first uses the Marcinkiewicz interpolation theorem to prove the boundedness of Tj,j ? Z with the integral domain limited to the interval[2j-1,2j).Then the author shows that T?,? is bounded from L?/1(R)× L1(R)to(?)(R)and also from L1(R)× L?/1(R)to(?)(R)via summing Tj up for all j ? Z to estimate T?.Next the author proves that T?:? is of restricted weak type(?/1,?,?),(?,?/?,?),(1,?,1-?),(?,1,?-?/?)via using some well-known results in weak LP spaces.Finally,by using the Marcinkiewicz interpolation theorem again,the author finishes the proof of the main result of this part.