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In this paper we consider operators with endpoint singularities generated by linear fractional Carleman shift in weighted H ölder spaces. Such operators play an important role in the study of algebras generated by the operators of singular integration and multiplication by function. For the considered operators, we obtained more precise relations between norms of integral operators with local singularities in weighted Lebesgue spaces and norms in weighted H ölder spaces, making use of previously obtained general results. We prove the boundedness of operators with linear fractional singularities.

The solvability theory of singular integral operators has developed independently in Hölder and Lebesgue spaces [

The norm in weighted Hölder spaces is defined in the following way. A function

is called Hölder function with exponent

Let J be a power function which has zeros at the endpoints

The functions that become Hölder functions and turn into zero at the endpoints, after being multiplied by

The norm in space

where

and

specifying that

We denote by

The norm of an operator

We denote a class of continuous functions on the segment

Let us introduce the following notation:

Let

Let

The norm in space

As we can see, the norms in spaces

By way of representatives of such types of operators we may consider the following operators with local singularities:

Such operators can be used in the study of boundedness, of belonging of some operators to Banach algebras and of the solvability of operators in weighted Hölder spaces, on the basis of known results for operators in weighted Lebesgue spaces.

It is useful to avoid two variables in the second term of the definition of the norm in Hölder spaces, for which we make use of

Lemma 1.

Let

then

where

On the basis of Lemma 1 the following theorem can be proved [

Theorem 1.

Let the following conditions hold for some operator

1) Operators

2) For any fixed

the following properties are fulfilled:

Moreover, inequalities

are correct.

It follows that operator

where

These results can be used in the study of operators in weighted Hölder spaces, on the basis of known results for operators in weighted Lebesgue spaces. In particular, operators with local endpoint singularities can be used in the construction of the left and the right regularizers in the study of Fredholmness of operators in weighted Hölder spaces.

We formulate a useful assertion which follows directly from Theorem 1.

Corollary 1. Let properties (1) and (2) be correct for the operator

Here

Then

where

We consider the operators

and

We note that for operators

Moreover, the following estimations hold

where

and

where

Theorem 2. Let an operator

and inequalities (2) be true.

If

then the operators

Proof. Let a function

We introduce functions

and

From the fact that

It follows that the function

is summable on segment

and

Condition (6) of the theorem makes it possible to choose constants

Now, we carry out an estimation of the expression

In doing so, we will use inequalities (5),

where

Here we have taken into account that

Since

when

From properties (5), condition (4) follows:

where

of Corollary 1 are fulfilled and we can apply it. Therefore operator

Since operator