We are concerned with finding a class of weight functions

In this paper, we are interested to find the best suitable function space for the weights

Let us first recall the classical Hardy inequality:

For

For

The most general sufficient condition (for any dimension) for the generalized Hardy-Sobolev inequalities is given by Maźja [

One such verifiable condition for

A general one dimensional weighted Hardy inequality has the following form:

Let

Having obtained

Let

The Moser-Trudinger embedding of

The rest of the paper is organised as follows. In Section

In this section, we recall the definition and some of the properties of symmetrization and certain inequalities concerning symmetrization that we use in the subsequent sections. For further details on symmetrization, we refer to the books [

Let

Then the

Next we give some inequalities concerning the distribution function and rearrangement of a function.

Let

The map

Finally, in the following proposition we state two important inequalities concerning Schwarz symmetrization (decreasing rearrangement).

Let

Next we state a necessary and sufficient condition for one dimensional weighted Hardy inequality due to Muckenhoupt (see 4.17, [

Let

In this section, we define the function space

Recall, for a bounded domain

Let

Let

This inclusion is strict as seen in the following example.

Let

For a bounded domain

Let

Let

In [

In the next proposition, we show that the weights considered in [

Let

Let

Next we verify that weights in

Let

Using Polya-Szegö inequality, we can easily verify that

In the next section, we see that the space

In this section, we give a proof for Theorem 1. Further, when

Let

The previous inequality is known for more general weights (even for measures), see [

For the simplicity, we let

Now by substituting back into (

Let

In the following theorem, we show that our condition is almost necessary for the generalized Hardy-Sobolev inequality.

Let

Let

Next we see how one can obtain Moser-Trudinger embedding and Hansson’s embedding using Theorem

From Theorem

Moser-Trudinger embedding: since

Hansson’s embedding: since

In this section, we give a proof of Theorem

First we prove the following compactness theorem.

Let

For

Since

Next we give an equivalent definition for the space

Let

Next we give example of functions in

We have already seen that

Whether all the admissible functions are in

For

The author thanks Professor Mythily Ramaswamy for her encouragement and useful discussions. He also thanks Prof. A. K. Nandakumaran and Prof. Prashanth K. Srinivasan for their continuous support and interest. The author also thanks the unknown referees for their suggestions that greatly improved this paper This work has been supported by UGC under Dr. D. S. Kothari postdoctoral fellowship scheme, Grant no. 4-2/2006 (BSR)/13-573/2011(BSR).