Gagliardo-Nirenberg Inequality as a Consequence of Pointwise Estimates for the Functions in Terms of Riesz Potential of Gradient

Our aim in this study is to give the Gagliardo-Nirenberg Inequality as a consequence of pointwise estimates for the function in terms of the Riesz potential of the gradient. Our aim here is to discuss boundedness of Reisz potential in term of maximal functions and to give the proof for Gagliardo-Nirenberg Inequality in term of Reisz potential. We will extend our result to discuss weak type estimate for Gagliaro-Nirenberg Sobolev inequality. Further, in this paper we are interested to extract Sobolev type inequality in terms of Riesz potentials for α is equal to one and to extend our work for weak type estimates when p is equal to one.